Post on 09-Nov-2020
Un schema aux volumes finis avec matrice signe pour les
systemes non homogenes
Slah Sahmim
To cite this version:
Slah Sahmim. Un schema aux volumes finis avec matrice signe pour les systemes non ho-mogenes. Mathematiques [math]. Universite Paris-Nord - Paris XIII, 2005. Francais. <tel-00010000>
HAL Id: tel-00010000
https://tel.archives-ouvertes.fr/tel-00010000
Submitted on 30 Aug 2005
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ËÖLFAyÌ$E®Ì ÊAL<AGHD6@6EU¿UFA=?A @LU<A<>E~IJ=CF)IHE$yÌ$L@LUÌ~D~@6GH|DGHz?AE$MA?6Ì Ê)$ÐL (SRNH) ¿1z?AGX $Ì$@|GHDE$F-MAE~?Ç´~D LU<>E~$¿)E_D(z?AG³L~DMA~ÅE$IH=C<A<aÖE$F-|EÖB&L6LFD(6?A@®IÃL<)@6=C<A@|GJ_DÑM L?D=C|GJGJIÃLU@6GHD6MAEfIÃL+6=CIH?)DGH=CFvMA?Ð<A@6=CB)IJÍ$E(MAEÀfGHE$ÐLFAFÓÌ$=@6@6E~6<a=CFAM&LUFzD®L?Çv|ÈX|D6Í$E$®F)=CFÐÊ)=CÐ=KCÍ$FAE~Ì$=CF)6GJM)$@6~ º
Ó©¬«¢ ©Z§~¢
ÔZL~D6ÊA=M)EMAE~fÅ=CIJ?)ÐE~*&F)GJE$DÑBAGHE$F LM&L<)D6$EL?Ç <A@|=CBAIHÍ$E$MAE$fIJ=GJ(MAEÌ$=CF)6E$@ÅUL DGH=CF º ÆF E! >E~D®Ì$E~D|DEÐ_DÊA=XMAEf6EÖB&LU6EÑ|?A@®IJL3=@6EGHFzD6$KC@6LIJEMA?´|ÈX|D6Í$E ~z?&LUDGH=CFA MAEBAGJIJLF E~Df<>E~@6E~D(IHE+Ì LIHÌ$?AI>MAEª6=CIH?)DGH=CFAM)GJ6Ì~=CFDGHFX?)E$M =C<>~@LUD6E$?A@|(FA=CFIJGHFA LUGJ@6E~$¿<>=C?)@MAE$´<A@6=CB)IJÍ$E$~Å=CIJ?ALFD M&LF)´MAE~ MA=CÐLGJF)E$Óz?AE$IHÌ$=CFAz?AE~ º ÖF<aE$?D @6E~´LU@6z?AE$@Ðz?AEÌ$E_D6DEv~D6ÊA=XMAEv<>E~?)D;_D@6EÅX?AEÓÌ~=CÐEV?)FAEvÐ_DÊA=XMAEM ~IJ$E$FD6&FAGJÑ=9IJE$3=CFAÌ~D6GJ=CF)DE~|DfÌ$= FAÌ~GJMAE~FzDL ÅCE~ÌIHE$ 3=CF)Ì~DGH=CFA(Ì$L@LÌ_D~@6GJDGHX?)E$MAEªÌ Ê&Lz?AEÌ$E~IJIJ?)IJE ºËÖLFAÌ$E Ì ÊAL<AGHD6@6EU¿L ÅULFDM)E´<A@|$6E~FDE$@VX?)E$IJz?AE~6Ì Ê)$ÐLVFz?A$@|GJz?AE$"#=CIJ?AE$ » GHFAGJ<>E~@6E~D|D LFDIÃL@|$|=CIJ?)D6GJ=CF-M)EÈX|DÍ~ÐE~ÊÈX<aE$@6Ba=CIHGJz?AE$(F)=CFÊ)=CÐ=KCÍ$FAE~$¿A=FÅULv@LU<A<>E~IJE~@IÃLvMA &FAGHD6GJ=CFM ?AFAEª|=CIJ?)D6GJ=CF L?|E$FAMAE$MAGH|D@|GJBA?DGJ=FALGJF)6G³z?AEIÃLÌ~=CFA|D6@6?AÌ_DGH=CFM)EIJLÐ_DÊA=XMAE+MAE~*#=IJ?AE » GHFAGJ º
m
«Z¤v©+©Z§~¢ ° «¢£_µ ¤1¢«y -£$+©Z§~¢ U§£~¤
ËÖLFA?AFpM)=C´LUGJFAE D ⊂ R¿ZIJEÓ|ÈXDÍ$E´ÊÈX<aE$@6Ba=CIHGJz?AEFA=CFpÊ)=CÐ=KCÍ$FAE<A@|E$FAM IJL 3=C@6E
6?AG,ÅULFDE e
∂U
∂t+∂F (U)
∂x= Q(x, t, U), ∀x ∈ D ⊂ R, t > 0
U(x, 0) = U0(x), ∀x ∈ D.
º
¾ÅE$Ì Q ?AFDE~@6E+6=C?A@|Ì$EB>=@6FA ºÖF<aE$?)DLIH=C@6®$FA=CF)Ì$E$@fIJEÖD6ÊA$=@6Í$E+6?)GHÅULFD e
lZn#:7Xopt % A)% /1 c U : D × R+ −→ R
#bd7: /1bd]_ 0/1b X7v]_*,1\ \^7 C1 ')+ ´/1 ] 76# J *J/1\*J7 \ª/1 J\('a/ '&/^\_ c 0/1b)\Ñ\_#X 01b& 7 \+\^/1b&%65$#X 01*J7_b& 7 \;N
U 7 \_ #Xbd7Ö\^/1*.# 0/1b X7"!$# %#Ð#-\^7_b)\;X7 \ J\_! "$# 0/1b)\"!|/1#-\^/1*.# 0/1bM~ "$*J7& c #ª /1#IX/ cbd7 R ⊂ D × [0,+∞[ N
∫
∂R
(Uηt + F (U)ηx) dσ =
∫
R
S(x, t, U)dxdt,
c c # U 7 \_ #Xbd7Ö\^/1*.#X 0/1bp]_*,1\ \_ 5$#&7;X7"!$# %#Ó* /(' U 7 \_ C1 $) U \$ J\ ~ c(*,] /1b, c 0/1b X7 b 1 cbd7" #X/1b& 0/1*J7v*J/1b X7 \´] /1#X$"_7 \ X7 J\^] /1b& 0"b&#X c % \ N
s[U ] = [F (U)].
+* , ©-¢ -¤ -¤.- ¢£$ ¤0/§~-§$
11 Q3246587:9<;6=>7@?$24BADC: FEHG7)62IADCJCK4¬N
,=C?)@(Ì$=CFAD@|?AGJ@|E+IÃL~DÊ)=MAEÑMAEÖÅC=CIH?AE$+&F)GJ$¿)=CF?)DGHIJGJ|EIJL93=C@6EÖGHFzD6$KC@6LIJEMAE+I $z?&L-DGH=CF º 6?A@IJE-M)=C´LUGJFAE R ⊂ D × [0,+∞[
º =CGHD η =
[ηx
ηt
]¿IJE-ÅCE~Ì~D6E$?A@ÐFA=C@6ÐLI
?AFAG,D LGH@6EÖE!ÇD6$@|GJE$?)@ ∂R ¿3@6=FzD6GJÍ~@6E+MA? MA=CÐLGJF)E R ºÆF?)D6GJIHGJLUFzDIÃL<A@|=C<>=6GHD6GJ=CF-GHG MA? D6ÊA$=@6Í$EÓ º º ¿A=CF L e∫
∂R
(Uηt + F (U)ηx) dσ =
∫
R
Q(x, t, U)dσ.
=
D " 6 $ I ÖF-MAGJ|Ì$@6_DGH6EÖIHEÑMA=CÐLGHFAEE~F-E$|<&LÌ$E+E_D(E$F-D6E$<A®E~F |?ABaMAGHÅXGJ6LFD D E~F-GJFDE~@|ÅULIJIHE$ 6?AÌ Ì$E~66G83 e=CF <>=C|E ∆xi = xi+ 1
2
− xi− 1
2
, i ∈ Z¿&E_D ∆tn = tn+1 − tn, n ∈ N
º
ÖFGHFzD6Í$KC@|E+I $z?&LUD6GJ=CF º ®6?A@?AFAEªÌ~E$IJIH?AIJE [xi− 1
2
, xi+ 1
2
] × [tn, tn+1]e
−∫ x
i+12
xi− 1
2
U(x, tn)dx+
∫ tn+1
tn
F (U(xi+ 1
2
, t))dt+
∫ xi+1
2
xi− 1
2
U(x, tn+1)dx−∫ tn+1
tn
F (U(xi− 1
2
, t))dt =
∫ tn+1
tn
∫ xi+1
2
xi− 1
2
Q(x, t, U)dx dt, º
E~Df=CFÌ ÊA=CGH6G,DMAEª<A@6E~FAMA@6E<a=C?A@(~DLUDfÐ=^ÈE$F³¿&IJLX?ALFDGHD6
Uni =
1
∆xi
∫ xi+1
2
xi− 1
2
U(x, tn)dx.
ÆF&F³¿h=CF<a=C6E
F (U(xi+ 1
2
, t)) = φ(Uni , U
ni+1).
º .
φ @6E~<A@6~6E$FD6EVIJE?A?Ç Fz?AÐ~@6GHX?)EI GHFzD6E$@ LÌ$E xi+ 1
2
¿&<>=C?)@ t Ì$=Ð<A@|GJE~FD@6E tn E~D tn+1
ºÔ ~X?ALUDGH=CF º ®MAE_ÅGHE$FD e
−∆xiUni + ∆tnφ(Un
i , Uni+1) + ∆xiU
n+1i − ∆tnφ(Un
i−1, Uni )
=
∫ tn+1
tn
∫ xi+1
2
xi− 1
2
Q(x, t, U)dx dt.
ÖF<aE$?)DLIH=C@6®$Ì$@|GJ@|EIHE+6Ì ÊA~´LÓ6=?AIÃL 3=C@6EÖ|?AGHÅULFDE eUn+1
i = Uni − rn,i
[φ(Un
i , Uni+1) − φ(Un
i−1, Uni )]+ ∆tnQ
ni
= =CF LÓ<a=C| rn,i =∆tn∆xi
º
QniE$|DI L<)<A@6= ÇGJÐLUDGH=CFfMA?+DE$@|ÐE|=C?A@|Ì$E¿$MA!&F)GJEGJÌ~GC<&L@ 1
∆xi∆tn
∫ tn+1
tn
∫ xi+1
2
xi− 1
2
Q(x, t, U)dxdt.
ÔZLMA_DE~@6GJF&LUD6GJ=CFMA?6?&?Ç Fz?A$@|GJz?AEE$|DÑIJE3=CFAM)E$E$FDf;~ÐEªMAEIÃLÓÐ_DÊA=XMAEªMAE~fÅ=IJ?AE$ AFAGJ º,yL@E!Ç)E~Ð<)IJE¿&<a=C?A@(IJLÌ$IJL6|EM)E$ Q 6Ì ÊA~´LU$¿&IJE?A?Ç Fz?AÐ~@6GHX?)E+ $Ì$@|GHD e
φ(Uni , U
ni+1) =
1
2(f(Un
i+1) + f(Uni )) − 1
2|Q(Un
i , Uni+1)|(Un
i+1 − Uni ).
Q E$DfIÃLv´L1D@6GHÌ$EÖÌ$L@LUÌ~D~@6GH|DGHz?AE+MA? Q− 6Ì Ê)$ÐL ºËÖLFAIHEÖ6Ì ÊA~´LvMAEªÀ=XE¿ Q(Uni , U
ni+1) = A(U) MA!AFAGHD(IJL´L1D@6GHÌ$EMAEªÀf=XE¿)= U @6E~<A@6
6E~FzD6EI _D LUD=^ÈCE~F vMA_DE$@|ÐGHFAE$@(E~F63=FAÌ~D6GJ=CFMAE~fM)E$?Ç _D LUD6 UniE_D Un
i+1
º
P
: D I D
Z ° «¢ ° ¢V¬ ° ¢ «¶£_¤ ° «y¢ £~µ ¤ -¢ -¢ ¢¯µ-¤£~¤ÂZ
Á =CFA|GJMA~@6=CFA®?AF |ÈXDÍ$E+MAEIJ=CGH M)EÖBAGJIJLFM&LFAR
m ¿GH6|?AEÖMAEIÃL$Ì$LFAGHX?)EÑMAE~+?&?AGHMAE$~¿E~Dfz?AG¬ $Ì~@6G,D e
∂W
∂t+∂F (W )
∂x= Q(x,W ) M&LFA R×]0, T [
W (x, 0) = W0(x).
º?>
=C?) LGJ|=CFA®I ÊÈX<a=DÊAÍ~6E3=CF)M&LE$FD LIHEÑz?AEÖIHEÖ<A@|=CBAIJÍ~ÐEMAE+ÀGJE~´LF)F´Ì$=@6@6E~6<a=CFAM&LUFzD$¿MA!AFAGZ<&L@ e
W0(x) = WL|G x < 0, E~D W0(x) = WR
|G x > 0,
LMAE~D?AF)E|=CIJ?DGJ=FL?)D=6GJGJIJLGJ@|EF)=D~E W (x, t) = H(xt
) º,IJ?A<A@|$Ì$GH6~ÐE~FzD eG =CFF)=DE aL
E_D aRMAE~+<>E~FzD6E$ªMAEvÌ~=C?A@6BaE$+M)EMAGJ|Ì$=CFD6GJFz?AGHD6=C?MAEvM)~DE~FDE¿¬GHIE!ÇGJ|D6E
?AF =C<>~@LUD6E$?A@FA=UD Rs D6E$I¬z?AE
W (x, t) = H(xt
)=
WL,|G xt< aL
H(xt
)= Rs
(xt,WL,WR
), 6G aL <
x
t< aR
WR,|G xt> aR.
º
ÀLU<A<>E~IJ=CF)z?AE¾IJÌ~@6?AMA=ÐE~DÖÄ®E$FAÉzÊ&LUIJMA=C?)F;=CFDÖ=CFD@|M&LUFA ¾ÖÄ) yz?AEVIJE$f$z?&L1DGJ=FAMAE6)LGHFD (#E~F&LFDÓL ÅCE~Ì?AF 3=CFAM E$F 3=C@|ÐE´MAE-´LU@6Ì ÊAEU¿yMA=CF)F&LGJE~FDÓIJGJE~? ;?AFAE-|=CIJ?DGJ=FL?)D6=C6GHÐGHIÃLGH@6E MA?Ð<)@6=CBAIHÍ$EMAEfÀfGHE$ÐLFAF³¿Cz?&LUFAM´IJLªM)GJ6Ì~=CFDGHFX?)GHD(GHFAGHD6GÃLIHE®MAEIJLª|=CIJ?DGJ=F~DLGHD|GHD?)$EvI E$FAMA@|=CGHDMAEªIÃLÐL@6Ì ÊAE º,yL@ªLUGJIJIHE$?A@|$¿aFA=C?A+E!Ç<A@6GHÐ=CF)Öz?AEÓFA=C?A+Ì~=CFA6GHMA$@|=CFAª?AFAEv|=CIJ?DGJ=FL?96E~FAªMAE$+MAGH|D6@6G8BA?)D6GJ=CFAMA?<A@|=CBAIHÍ$E º > _¿)E$F ?)DGHIJGJ6LFD(IÃL<)@6=C<a=C6G,DGH=CF ii) MA?-DÊA~=C@6Í~ÐE º º e
∫
∂R
[W.nt + f (W ) .nx] dσ =
∫
R
Q (x,W ) dxdt. º =
ÖF|E-<A@6=<>=C|E-MAE-Ì$=FA|D6@6?AGH@6E ?AF |Ì ÊA$ÐLFX?)Ð~@6GJz?AE-BAL6-?AF)GJz?AE$E$FDÓ|?A@vIJE~MAE$?XÇ<A@6=<A@6GH~D~ º ®E~D º = º
@
D " 6 $ I
ËÑ!AFAGJ|6=CFAf<a=C?A@?AF ´LGHIJIJLKCE?AFAG83=C@6E ern =
∆tn∆x
, L ÅCE$Ì ∆tn = tn+1 − tnE_D ∆x = xi+ 1
2
− xi− 1
2
¿)IHE$<&LUMAEÑD6E$<AE~DM E~6<&LUÌ$E@6E~6<aE$Ì~D6GHÅE$E$FD º=C?)fF)=DE~@6=CFA xi
IJE+Ì~E$FD@|EM)EIJLvÌ$E$IHIJ?AIHE ]xi− 1
2
, xi+ 1
2
[º
GJFA|<AGJ@6LFDMAEIJLÖMA$ÐL@|Ì ÊAEM)EÑÏ+=XMA?AFA=^ŠϪ=XMhORz¿XÏ+=M =hE~D E~FÐ?)D6GJIHGJLUFzD ?AFAE<A@6E~ÐGHÍ$@|E3=CGJ IJE~<A@6=<A@6GH~D~ º E~D º = _¿X6?A@®IJEÑM)=C´LUGJFAE R =]xi− 1
2
, xi+ 1
2
[×[tn, tn+1[¿=F-$Ì$@|GHD e
W n+1i = W n
i − rn
[F(Rs(0,W n
i ,Wni+1
)− F
(Rs(0,W n
i−1,Wni
)]+ ∆tnQ
ni ,
= QniE~|Df?AFAEL<A<)@6= ÇGJÐLUDGH=CF MAE 1
∆tn∆x
∫
R
Q (x,W ) dxdt.
,=C?)@I _ÅULIJ?&L1DGJ=FÖMAE F (Rs(0,W ni ,W
ni+1)) ¿ I GJMA~EGJÌ~GE$DyM _ÅG,DE~@¬MAEÌ LIHÌ$?AIHE$@¬IÃL|=CIJ?DGJ=F
E_Ç)LÌ_DE Rs ¿C=C?v<;$EM GJFD@|=M)?AGJ@|E?AFÓ<A@6=BAIJÍ~ÐEM)EÀfGHE$ÐLFAFIHGJFA$L@6GH6 º =?AÌ ÊAE$@|Ì ÊA=CFAMAGJ@|E$Ì_DE$E$FD?AF)EÑÅULIJE~?A@(L<A<)@6=XÌ ÊA$E+MAE+IJLV6=IJ?)D6GJ=CFÐE_Ç)LÌ~D6E¿&E~F-E_Ç<AIJ=GHD LUFzD®IJE~<A@6=<A@6GH~D~ º ®E_D º = º¾ÑGHFA6G¿A=CF$Ì~@6GHD<AL@E_ÇE$<AIJE+|?A@IJE+Ba=C@6M xi+ 1
2
e
W ni+ 1
2
= L<A<A@|=^ÇGH´LUD6GJ=CF´MAE Rs(0,W n
i ,Wni+1
).
,=C?)@V=CB)D6E$FAGH@VÌ~E~D6D6E L<)<A@6= ÇGJÐLUDGH=CF³¿¬=CFR?)DGHIJGH6EÓMAE FA=?)ÅCE$L? IÃL<A@6=<A@6GH~D º = +6?A@?AF=C?)ÅE$@|D πθ =]X−, X+[×]tn, tn + θ[ DE~I¬X?)E xi+ 1
2
∈]X−, X+[º
ÆF|?A<A<a=CLFDz?AE X−, X+ E_D θ > 0 |=CFDfÌ ÊA=CGJ|GJ(MAE+D6E$IJIHE%L½~=CFz?AE ∀t ∈]tn, tn + θ[ e
Rs
(X− − xi+ 1
2
t− tn,W n
i ,Wni+1
)= W n
iE~D Rs
(X+ − xi+ 1
2
t− tn,W n
i ,Wni+1
)= W n
i+1,
E~DfE$F GJFD6@6=XMA?AGJ6LFDIJE~(MAGJD LFAÌ~E$ e dX− = |X− − xi+ 1
2
| E~D dX+ = |X+ − xi+ 1
2
| ¿=CFL
X+∫
X−
W (x, tn + θ) dx = dX−W ni +dX+W n
i+1−θ[F(W n
i+1
)− F (W n
i )]+
∫
πθ
Q (x,W ) dxdt.
ÖFMA &FAG,D e
R
: D I D
W ni+ 1
2
=1
dX− + dX+
X+∫
X−
W (x, tn + θ) dx.
¾ÅE$̪IJEÌ ÊA=CG Ç e X− = xiE_D X+ = xi+1
¿=CF =CB)D6GJE~FzD®?AFAE+<A@|E$GJÍ~@6EÑE!Ç<A@6E~66GH=CFKC$F)$@LUIJE¿E$F 3=FAÌ~D6GJ=CFMAE θ ¿>MAEVIÃLÓ6=CIH?)DGH=CF LU<A<A@6=XÌ ÊA~E W n
i+ 1
2
GJ|6?AEM)EVÌ$E_D6D6EV<A@|E$GJÍ$@|E_D L<aE¿>z?AEFA=C?AfL<)<>E~IJIJE~@6=CF)(~D LU<>EªMAEª<A@|$MAGHÌ~DGH=CF e
W ni+ 1
2
=1
2(W n
i +W ni+1) −
θ
∆x
[F (W n
i+1) − F (W ni )]+ θQn
i+ 1
2
, º P
=
Qni+ 1
2
= Q(xi, xi+1,W
ni ,W
ni+1
)
E$D?)FAEVLU<A<A@6= ÇGJÐLUD6GJ=CF³¿5vMA!AFAGJ@MAEªÐLFAGJÍ~@6E |?)MAGJÌ~GJE$?)6E¿&M)E
1
θ∆x
∫
πθ
Q (x,W ) dxdt.
F-<A@|E$GJE~@®Ì ÊA=CG ÇÐMA?-<&L@6LÐÍ_D@|E θ LV_DÖM)EÖIJE<A@6E~FAMA@6EÑÌ$=CÐE?AFAE%3@6LÌ~D6GJ=CFÐMA?-<&L®MAEDE~Ð<A ∆tn
¿&E$F $Ì$@|GHÅULFD θ =αn
i+ 1
2
2∆tn
º
ÖFWLU@6@6G,ÅCELGHFA6G ;IÃL;<A@|E$GJÍ$@|E 3=C@6E MA? |Ì ÊA$ÐL À Î =CI,ÅCE~?A@vMAE ÀfGHE$ÐLFAF f=CFÎ =C=CKCÍ~FAE e
W ni+ 1
2
=1
2(W n
i +W ni+1) −
αni+ 1
2
2rn
[F (W n
i+1) − F (W ni )]+αn
i+ 1
2
2∆tQn
i+ 1
2
W n+1i = W n
i − rn
[F(W n
i+ 1
2
)− F
(W n
i− 1
2
)]+ ∆tQn
i .
º @
Á E~D|DED3=C@|ÐE MA?6Ì Ê)$ÐL )À Î Lp_D;?)D6GJIHGJ6L ÅCE~Ì6?AÌ~Ì$$<a=C?A@ÐMAE$ <)@6=CBAIHÍ$E$ÐFA=CFÊA=C=CKCÍ~FAE$ª^ËjM&LFAIHE+Ì LMAEªÐLGJIHIÃLKCE~ ?)FAG3=C@|ÐE~ 3Å=CGJ@ Ä®E~F& ºD tXopi 1r³sZt Q CtL i k´xtwi 1iAoH7kmt Zt Q kaxt αn
i+ 1
2
IÁ EÖ<&L@6LÍ~D@|EÖLV<a=C?A@ =CBX|E~Ì~DG8¬MAEÖÌ~=CFD@TSCIJE~@®IÃLMAG >?A|GJ=CFÐFX?)Ð~@6GJz?AEMA?-6Ì Ê)$ÐLÌ$=CÐEIJE+=CFD@|E+I LF&LI,È|E+ÐE~FA$EªM&LUFAIJEÖÌ$L6Ì LUIÃLGH@6EÖM&LUFAIJE+Ì Ê&L<)GHD@|E 2
º ,L@E_ÇE$<AIJE+MALFAIJEÌ LM ?AFAE´~z?&LUDGH=CFp|Ì LIJLGJ@|EÓIJGJF) LGH@6EÓ6?A@ÐLGJIHIÃLKCEV?AFAG3=@6EIHEÐÌ Ê)=CG,Ç MAE αn
i+ 1
2
= 1
D " 6 $ I
@6E~MA=CFAFAEE_Ç)LÌ_DE~ÐE~FzDfIHE+6Ì ÊA~´LMAEÔLmÇpE~FAMA@|= º
K-sZk+1t'Ai 1q3i k^t suUlZnXopi B D FG IËÖLFA IJE(BA?D®MAEfI E_ÇXD6E$FA|GJ=CFMA?´|Ì ÊA$ÐL À Î L?Ì LU BAGHMAGJE$F)6GJ=FAFAE$I¿=GJI&E~|D MAG Ì~GJIJEMAEªMA!AFAGJ@?)F ~z?AGHÅULIHE$FDMAE+IÃLM)GJ|DLFAÌ~E ∆x ¿&GHIdE~|DfL<A<AL@6?FA~Ì$E$|LGH@6EªMAEª<A@|=C<>=6E$@(?)FAEÅUL@6GJLFDE®M)E À Î F)ELGHLFD <AL L<)<&L@LJD@|E Ì~E~D|DEMAGH|DLFAÌ$E M&LFAIÃLf<AÊ&L|E<A@6~MAGJÌ_DE~?A@MA? 6Ì ÊA~´L º
Á E$IJLRÌ$=FA6GH|DE R~Ì$@|GJ@6E θ = αni+ 1
2
θ = θ E$D´M)!&FAGf<&L@IÃLÅXGHD6E$6|E MAEÀ?ALF)=mÅ cÅC=CGH@&KC?A@|E º Ì$=ÐEÖ6?)GHD e
xX+xi+ 1
2
X−tn
tn+θ
tn+θ
t
W ni W n
i+1
º
θ =∆x
2Sni+ 1
2
.
Á E$IJLLUÐÍ~FAEÓIJLFA=C?)ÅE$IHIJEªE_Ç<A@|E$6|GJ=CF MAEªIÃLv<AÊAL6Eª<A@|$MAGHÌ~DE~?A@MA? 6Ì ÊA~´L e
W ni+ 1
2
=1
2(W n
i +W ni+1) −
αni+ 1
2
2Sni+ 1
2
[F (W n
i+1) − F (W ni )]+αn
i+ 1
2
2
∆x
Sni+ 1
2
Qni+ 1
2
. º R
D tXopi 1r³sZt I ,=C?)@ÑIJE~<A@6=CB)IJÍ$E$D@6LGHD6$M&LF)ÑÌ$ED@6L^ÅULGHI0¿IJEªDE$@|ÐEFA=CFÊA=C=CKCÍ~FAE QÌ$=CFD6GJE$FD KC$F)$@LUIJE$E$FD ?AFÓDE~@6EMAEMA~@6G,ÅC$E6<&L1DGÃLUIJEM =C@|MA@6E º ¾ÑGHFA6G¿<>=?A@IJE(|ÈXDÍ$E
: D I D
MAE6)LGHFD (#E~F&LFD ¿yIÃL <A@|E$GJÍ~@6E Ì~=CÐ<a=C6LFDE-MA? DE~@6E 6=?A@6Ì~E E~|DÓFz?AIHIJE E_DvIÃL;|E$Ì$=FAMAE $Ì~@6GHD Q = −gH dZ
dxMA=CFD´?AFAEL<)<A@6= ÇGJÐLUDGH=CF |DLFAM&L@|M<a=C?A@IÃLR<)Ê&L6E<A@6~MAGJÌ_DE~?A@
$Ì~@6GHDQn
i+ 1
2
= −gHni +Hn
i+1
2
Zi+1 − Zi
∆x.
ÔZLÖMAGJD LFAÌ~E ∆x M)GJ6<AL@LHDyM)=CFAÌMA?D6E$@|ÐE|=C?A@|Ì$E(MAEI _D L<aE<A@|$MAGHÌ~D6E$?A@~¿E~D <>E~@6E~DLGHFA6GI E_ÇXDE~FA6GH=CF MA?|Ì ÊA$ÐLL?Ì LBAGHMAGJE$FA|GJ=CF)FAE$I ºÔZL3=@6E+MA!&F)GHDG,ÅCEªMA? 6Ì ÊA~´L)À* Î M&LUFAIHEÖÌ LUKC~FA$@6LI³ ~Ì$@6G,DfMA=CFAÌ e
W ni+ 1
2
=1
2(W n
i +W ni+1) −
αni+ 1
2
2Sni+ 1
2
[F (W n
i+1) − F (W ni )]+αn
i+ 1
2
2
∆x
Sni+ 1
2
Qni+ 1
2
W n+1i = W n
i − rn
[F(W n
i+ 1
2
)− F
(W n
i− 1
2
)]+ ∆tQn
i . º ^
C
¾ÑÄ) »®º ¾ÑIHÌ$@|?AMA= LUFAM »®º Ä®E$F)ÉXÊALIJMA=?AF³¿ JzX]_ \^/1*.#X 0/1b)\ / c2&7 07 b&b 'a/"$*J7 /c2&7\ 2)*c*J/!1! 7_-765$#A 0/1b)\)! c32 9"_/1 / \_ 7 ' ¿ Á =C<A?)D º > » IJ?AGHMA !C) !¿)FA= º =)¿ = > .@ =(P º
Ä E$F& »®º Ä E$FAÉzÊ&LIHMA=C?AFd¿ ba*6^\_ J\b,21*. 7 0/1b/pbd7! b& c 7A/1*.# ´7-\^] 2&7 ´7 /1bd/1b)2&/ ´/3X7_bd7 /1#z\\^\_ 7 V\ ¿ » # Á ¾.)¿ Î ,>¿(À º(Î E~@6BAGHF³¿Ë º Ò+@39FAE$@ ÆMA !¿,@6=XÌ$E~E$MAGHFAKCÖ=DÊAE &(Ê)GJ@6M FzD6E$@|F&LUDGH=CF&LI$XÈXÐ<)6=CGH?A =CF » GJF)GHDE#=CIH?AE$3=@Á =C<AIHE_Ç-¾<A<AIHGJÌ L1DGJ=F³¿AUC¿A<)< º O=QR5QPO= º
`Ï+=XMhOR" º Ò º Ï+=MA?)FA=^Å>¿G Ö7_ 7_bd] 7 ´7_32&/' /1ªb&#H ´7_! 0]6*y]6*J]_#X*, 0/1bp/; J\^] /1b& 0"b&#&/1#z\ \^/1*.# 0/1b)\Ñ/®32&7765$#A 0/1b)\Ñ/ 2?/'1ba v 0] \ ¿ ¹ LUD º B º º º ROR !¿&QPX5.CQ= º
`Ï+=XM =C ¿ Ö7_7_bd] 7 ´7_c2&/' /1Rc2&7 ]6*J]_#*, 0/1b /\ 2&/^] ! 7 \ ¿¾ÑE$@ º¹ L1DÊ º =XÌ º &@6LFA6I º 0 % RQ= _¿@.O@QR55.QR º
.
>
:
W W
* W W 1D
±Z
ËÖLFA Ì$EÖÌ Ê&LU<AGHD6@6E¿XFA=C?)<A@6~6E~FzD6=CFA(?AF)E+LF&LI,ÈX6EÑÐLUD6ÊA$ÐLUDGHz?AEMA?-6Ì ÊA~´L )À ÎM&LFAIHEfÌ$LÊA=C=CKCÍ~FAE 1D
ºÁ E~D6D6EÑLUF&LIHÈX|EfF)=C?A L<aE$@|ÐGH MAEMA_DE$@|ÐGHFAE$@?AF)EÅULIHE$?A@=<5DGH´LIHE®M ?AFÐ<&L@LUÐÍ_D@6E(M)EfÌ~=CFD@TSCIJE(MAEIJLªM)G >?A6GH=CFvGJFDE~@|ÅE$F&LFD M&LFA IÃLÖ<)@6E$GJÍ~@6E(_D L<aEMA?p|Ì ÊA$ÐL ºÁ E$IJL F)=C?ALÌ$=FAMA?AG,D9?AFAEFA=C?ÅCE$IHIJEÓE_Ç<A@|E$|6GJ=F MA?p|Ì ÊA$ÐL (SRNHS) ¿LGJ6LFDÑGHFDE$@ÅCE~FAGJ@IÃLÐÐLUD6@6GJÌ~E|GJKCF)EM ?AF=C<a$@6LUDE~?A@ÑM)EVÌ$=FzÅE$Ì_DGJ=F9L?;FAG,ÅCE LU?;MAEÌ~E~D6D6E;~ÐE+<)@6E$GJÍ~@6E+_D L<aE º
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ÔZLM)G Ì$?AI,DvMAEÓIJL@|$|=CIJ?)D6GJ=CF9LUF&LIHÈzD6GJz?AEMAEÓÌ~E$@|DLGJF)ÈX|DÍ~ÐE~ÊÈX<aE$@6Ba=CIHGJz?AE$ªÊA=Ð=KCÍ$F)E$FA=CFÓIHGJFA$LGJ@|E$=C?vMAE(IJ=GJyM)EÌ$=CF)6E$@ÅULUDGH=CFÓLÖGJFAÌ~GHD6®<AIH?A6GHE$?A@| Ì ÊAE$@|Ì ÊAE$?)@6ÖMA~ÅE$IJ=<5<>E~@®MAE~6Ì Ê)$ÐLMAEfÅC=CIH?AÐE~AFAGJ º Ô¬EÑ<)@6E$GJE~@ |Ì ÊA$ÐLVMAEÌ$EKCE$FA@|E+L_DMA~ÅE$IJ=<A<>ÑE$F1959 <&L@ Ϫ=XMA?AF)=mŠϪ=XMhORX¿Ï+=M =)X?)GA L<A<A?)GJE®|?A@IÃLf@6~6=CIH?)DGH=CFE_Ç)LÌ_DEM)?v<A@6=CB)IJÍ$EMAEÀGJE$ÐLFAFd¿UE_DX?)GAE$DÖI =C@6GHKCGJFAE MAE(IJLÑ<AIJ?)<&L@|D MAE$~D6ÊA=M)E$ MAG,DE~MA$Ì~E$FD@|$E$ º ,yL@IJL6?AG,DEU¿MAG >$@|E$FD6Ì Ê)$ÐLMAE(ÅC=CIH?AÐE~$&FAGH |E6?AÌ~Ì$Í$M)E$FD º ÖFF)=DE<&L@E!ÇE$<AIJEÌ$E~IJ?AG)MAEÔZL1Ç7 » @6GHE$MA@|GJÌ ÊA Ô =C>X?)G>E~|D®D6@6Í~ MAG8 d?)6GdE~D®FAE<aE$@|ÐE_D®<AL®IJL|GJ?AIJLUDGH=CFvFz?A$@6GHz?AEMAE$+<)@6=CBAIHÍ$E$E$F9MAGJE$F)6GJ=F MAE~?ÇM E$|<&LÌ~E|?A@ÖÐLGJIHIÃLKE+FA=CF|D6@6?AÌ_D?A@|¿¬LUGJFA|Gz?AEvIHE6Ì ÊA~´LMAE À?ALUFA=^Å Àf?) =) z?AG KL@6M)EÐIHEÐÌ$L@LÌ_DÍ~@6E´M)G >?A6G8ÐLGJLMAE~D?AFAEÐE_ÇXDE~F56GH=CFRE~FRM)GJE$FA|GJ=CF9MAE~?Ç M E$6<ALÌ$E º ÆF 1981 ¿ZÀf=XEÓL<A@6=C<a=C|?)Fp6=IHÅCE~?A@ªMAEÐÀfGHE$ÐLFAFL<A<A@|=Ì Ê) IHGJFA$LGJ@|E À=E@)f<aE$@6E~D|D LFDÐIÃL6GH?AIJLUDGH=CF MAE$Ð$z?&LUD6GJ=CF)ÐM Æ?AIJE~@ÓMAE IJLM)ÈXF&LGJz?AEMAE$KCL º ÆF 1999 ¿ÏLUIJIJ=?"!~DE_D ¹ L|E$IHIÃL ¹ » ÏERQR>=CFD<A@|=C<a=C6?AF FA=C?ÅCE LU?6Ì ÊA~´LRFX?)Ð~@6GJz?AEU¿ # » Àf=XE¿ M)ED ÈX<>EÅC=IJ?AE$ &FAGH<a=C?A@IJE~ÐÈX|DÍ~ÐE~´M $z?&L1DGJ=FA
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ÊÈ<aE$@|B>=IJGJz?AE~Ñ?AF)GJMAGHÐE~FA6GH=CFAFAE~IJIJE~(E~DÑBAGJMAGHÐE~FA6GH=CFAFAE~IJIHE$(FA=CF IJGJF) LGH@6E$ÊA=Ð=CKÍ$FAE~$¿aE$F6E BALLFDv|?A@VIJL~D6ÊA=M)E´MAEÏ+=XMA?AFA=^ÅRL<A@|Í$IJLIJGHFA LU@6GJ6LUDGH=CFMA? <A@|=CBAIHÍ$E¿ZE$FA|?AGHD6EGJI6E~@L´<A@|=CIJ=FAKC<>=C?)@ÑIJE~ÑÈDÍ~ÐE~+FA=CF;Ì~=CFA6E~@|ÅULUD6G3 Ä(Ï Î CX¿³Ï Î ).)¿³Ï Î º ,IH?A@6~Ì$E$E$FD )º)¹Rº ϪÊAGHM&LKCIHGÃL)¿U¾ º Ò+?AB&L@|=E_DÏ º ÔZE Á = ÏÒªÔ Á RQ=d=CFD®MA~ÅE$IJ=<A<>?AFFA=C?)ÅE L? 6Ì Ê)$ÐLM)EVÅC=CIH?AE$&FAGHÖz?AG L<)<A?AGJEV6?A@+IJL´FA=D6GJ=CFMAE"?&?Ç9Ì$L@LÌ_D~@6GJDGHX?)EE~Dªz?AGyL MA=CFAFAvM)EB>=FA+@6~6?AI,D LUD6ÖM&LFAÑIJE$ÑÌ L 1D E~D 2D <>=C?)@ÖIJE~Ö|ÈX|D6Í$E$ªFA=CFÊ)=Ð=KCÍ$FAE~ º,yL@ÖLGJIHIJE$?)@6$¿hIJL´FA$Ì~E$|6GHD6ÓMAEMA_ÅCE$IH=C<A<aE$@+MAE~+6Ì ÊA~´LU+MAEVÅC=IJ?AE$%AFAGJÖ<)IJ?AÖ<)@6$Ì~GJÖE_DE Ì LUÌ$E$L´Ì~=CFAMA?)GHDÐIÃL´@|E$Ì ÊAE~@6Ì ÊAEÓMAE~Ö6Ì ÊA~´LU+M =@6MA@|E6?A<a$@|GJE~?A@ÑDE~IJX?)EvIJEV|Ì ÊA$ÐLMAE;ÔZL1Ç7pE~FAMA@6= :z?AGfE$D ?AFW6Ì ÊA~´L D@|=CGJ<a=CGJFD~¿MAE |E$Ì~=CFAM=C@|MA@6E E$FE~6<&LUÌ$E E_DE$F D6E$<A$¿hIHGJFA$LGJ@|E$E$FD L2 |DLBAIHE ºa»®º ËÑE#?)ÈX|D Ë #+ > LÓMA~ÅE$IH=C<A<aV?AF6Ì ÊA~´L L ÅE$Ì?AF ?A?Ç Fz?AÐ~@6GHX?)EÊÈXBA@|GJMAE$Ì~@6GHDÌ~=CÐEv?AFAEÌ$=BAGJFALGJ|=CF9Ì$=CFÅCE!ÇE´MA? ?&?Ç MAEÐÔZL1Ç7pE~FAMA@|= 9MAEªMAE$?XÇ)GHÍ$Eª=C@6MA@|EªE~DMA? ?&?Ç MAEÔZL1Ç7 » @|GJE$M)@6GJÌ Ê)MA? <A@6E~ÐGHE$@(=C@|MA@6E ºÖFF)=DE~KLIJE~ÐE~FD¬IJE M)~ÅCE~IJ=C<)<>E~ÐE~FzDMAE~¬6Ì ÊA~´LUM)E ÅC=CIH?AÐE~A&FAGH³MAED ÈX<>E<A@|$MAGHÌ~D6E$?A@Ì$=C@|@6E~Ì~DE~?A@VÌ~=CÐEvÌ~E$IJ?)GMAE´ÀGJÌ ÊD6VÈCE~@E_DVÔZL1Ç7pE~FAMA@6= À ¹ =(P¿ZÌ$E~IJ?AGMAE ¹ LÌ Á =C@´LUÌ É Á =C@ =OR³E_D(IJEf6Ì Ê)$ÐL ;¾ » pE~GJKCÊDE~M-¾ÅE$@6LKCE » IH?Ç GHFzD6@6=XMA?AG,D <AL@(Æ ºA»®º &=C@6=A¿z?AG³KCE~FA$@6LIJGH6EÖIHEÑ6Ì Ê)$ÐLÓM)E+ÔZL1Ç7 RE$FAMA@|= ¿AIJE6Ì ÊA~´LvMAEÏ+=M)?AFA=^Å´<a=C?A@®IJE$®|ÈX|D6Í$E$MAE IJ=CGMAE Ì$=FA6E~@|ÅULUDGH=CFVF)=CFVIHGJFA$LGJ@|E E~DIJE 6Ì ÊA~´LfMAE L@|ÐGHFAK Ä®E L &=C@VRQR¿^Ì$E®|Ì ÊA$ÐLE$DM =C@|MA@6EvM)E$?Ç9E~F9E$6<ALÌ$EÓE~DªE~FDE$<A+E_Dª<>E~@6E~DªMAEvIHGJGHDE~@IJE$+=6Ì$GHIJIJLUDGH=CFAfFX?)Ð @6GHX?)E$$¿Ì~E-6Ì ÊA~´L L~D6L<A<AIHGJz?AEMAE~<)@6=CBAIHÍ$E$E~FWM)GJE$FA|GJ=CFAVMAE$?Ç E~DvD@|=CGJVE$FE$|<&LÌ$E Ä&CRQ@A¿hÄ+&CR(P¿ &=C@ RC0¿h=C? E$F)Ì$=C@|EVÌ$E~IJ?AGZMAEÄ®E~FAÉzÊ&LIJM)=C?AF Ä®E~F&B&L|V|?A@?)FAE@6E~´L@|z?AE M L?D=C|GJGJIÃLU@6GHD6M)?<A@|=CBAIHÍ$E M)E ÀGJE~´LUFAFªMAE IJL(6=CIH?)DGH=CF+M ?AFV<A@|=CBAIHÍ$EFA=CFÊA=C=CKCÍ~FAE º ËÖLF)®Ì$EÌ Ê&L<AG,D@|E¿zE$FÐ6EÑBALLFD®6?A@ IJE$ <A@|=C<A@6GH~D6$ MAE$ 6Ì ÊA~´L MAEÅ=CIJ?)ÐE~&FAGHÑ<a=C?A@ÑIHE$f|ÈX|D6Í$E$+ÊÈX<>E~@6Ba=CIJGHz?AE$ÊA=C=CKCÍ$F)E$ ϪÀ R)U¿³ÏªÀ RQ=FA=C?)Ö<A@6=<>=C|=CFAMAELGJ@|EI LUF&LIHÈX|EVÐLUDÊA~´L1DGJz?AEªMA?|Ì ÊA$ÐL )À* Î Ä®E~F&¿dMALFAÑIHEVÌ$LÖMAE~Ö<A@6=BAIJÍ~ÐE~ÊA=C=CKCÍ~FAE$6Ì$LIÃLGH@6EE_DÅE$Ì_D=C@|GJE$IzE_DyM E_Ç<AIHGJÌ~GHDE~@¬IJEÌ ÊA=CG ÇMA?<&L@LUÐÍ_D@6EMAE Ì$=FzD6@VSCIHE αn
i+ 1
2
º
£!² ¬¤ Z- ¦£~¤Â¨Zª£$ª§_«¤
ÖF Ì$=CFA|GJMAÍ~@6E-IHE |ÈX|D6Í$E-@6~KCG<&L@V?AFAE-IJ=G MAE-Ì~=CFA|E$@|ÅULUD6GJ=CF Ì E~|D<;MAGH@6E IHE´<A@|=CBAIHÍ$EÊÈ<aE$@|B>=IJGJz?AEª|Ì LIJLGJ@|EÖÊA=C=CKCÍ~FAE 1D 6?)GHÅULFD e
∂w
∂t+∂f(w)
∂x= 0, ∀ (x, t) ∈ R×]0,+∞[,
w(x, 0) = w0(x), ∀ x ∈ R.
º
=C?) FA=?A<A@6=C<a=C|=CFAMAE2LGJ@|E(?AFAEfLF&LI,ÈX6E´L1DÊA~´LUD6GJz?AE(<a=C?A@IHEf|Ì ÊA$ÐLªMAEfÅ=CIJ?)ÐE~&FAGH )À* Î ~Ì$@6G,DV<a=C?A@ªIJL|GJ?)IÃLUD6GJ=CFFz?A$@|GJz?AEMAE$<A@6=BAIJÍ~ÐE~F)=CFpÊA=Ð=CKÍ$FAE~<&LU@»®º Ä®E~FAÉzÊ&LIHMA=C?AF Ä E$F&+E~D MA &FAG+M&LF) IHE;<A@|E$GJE$@-Ì ÊAL<AGHD6@6E º ,=C?)@ M)E$@LGH6=CF)-MAE6GHÐ<AIHGJÌ~GHDU¿ FA=C?) Ì$=CF)6GJM)$@6=FA IJE Ì LU ÊA=C=CKCÍ~FAE º®Á E;6Ì Ê)$ÐL $Ì$@|GHD´<>=C?)@´IJE |ÈXDÍ$E
O=
DDI D 1D
0 º ®MAEªIJLÓÐLFAGHÍ$@|EÖ6?AG,ÅULFDE e
wni+ 1
2
=1
2
(wn
i+1 + wni
)−
αni+ 1
2
2Sni+ 1
2
(f(wn
i+1) − f(wni )),
wn+1i = wn
i − r(f(wn
i+ 1
2
) − f(wni− 1
2
)),
º
= αni+ 1
2
E$|D?)F<AL@LÍ~D6@6EM)E(Ì$=CFD@TSCIJE®E~D Sni+ 1
2
E~|DIÃLfÅG,DE~66EIH=Ì$LIJE MAEÀ?ALF)=mÅMA=CFAFA~E<&L@
Sni+ 1
2
= max(∣∣f ′(wn
i+1)∣∣ , |f ′(wn
i )|), r =
∆t
∆x,
= ∆t E$DIHE+<&LMAE+DE~Ð<)fE_D ∆x E$DIHE+<&LM E$|<&LÌ$E ºËÖLFAÌ$E_D6DE<&L@DGHE¿&FA=C?)ÑLIHIJ=CF)+LGJ@|EªI ~D?)MAEª´LUD6ÊA$ÐLUD6GJz?AE+MAEÌ$E6Ì ÊA~´LM&LFAfIJEªÌ$LIJGHFA LGH@6EM&LF)?AF <A@|E$GJE~@ DE~Ð<)®E~D®M&LFA®IJEÑÌ$L®F)=CF-IJGHFA LGH@6EM&LF)®?AF-MAE~?ÇGJÍ~ÐEfDE$<A~¿E$FKL@|M&LFD αn
i+ 1
2
= α(wn
i , wni+1
) ¿ E~D=CF|E<A@6=<>=C|E;MAE;<)@6$Ì~GJ|E$@-IÃL ÅLUIJE$?)@´MAE αni+ 1
2<>E~@6E~D|D LFDfMAEªÌ$=CF53$@6E~@MAEªB>=CF)FAE$<A@|=C<A@6GH~D6$fL?6Ì ÊA~´L º
1 1 Q 5 $?$4DG¬?&9<C
ÖF<A@|E$FAM f(w) = c w ¿AL^ÅE$Ì c ∈ Rº Ô¬Eª6Ì ÊA~´LÓMAE+ÅC=CIH?AE$'&FAGH( $Ì$@|GHDLIH=C@6
wni+ 1
2
=1
2
(wn
i+1 + wni
)−αn
i+ 1
2
2sgn(c)
(wn
i+1 − wni
),
wn+1i = wn
i − rc(wn
i+ 1
2
− wni− 1
2
).
º .
Ô¬E |Ì ÊA$ÐLM)!&FAGE~F º . _¿ E~|DÌ$=CFA|E$@ÅL1DGE~DÌ~=CFA6GH|DLFD<AL@ZÌ$=CFAD@|?AÌ~D6GJ=CF³¿^E_DIÃL+3=CFAÌ_DGH=CFMA? ?&?ÇFz?A$@|GJz?AEªE$DM)=CFAFA~E<&LU@
g(wn
i , wni+1
)= f
(wn
i+ 1
2
)= c wn
i+ 1
2
.
ÔZL3=CF)Ì~DGH=CF MA?C?A?ÇFz?AÐ~@6GHX?)E g E$D´Ì~=CFDGJFz?AEU¿ ÔZGH<A6Ì Ê)GHD ~GJE~FAFAE<;E~FWE dE_D´<a=C?A@vD=?)Du ∈ R
=CF L
∣∣g(wni , w
ni+1) − f(u)
∣∣ =|c|2
∣∣∣(wn
i+1 + wni
)− αn
i+ 1
2
sgn(c)(wn
i+1 − wni
)− 2u
∣∣∣
=|c|2
∣∣∣(1 − sgn(c)αn
i+ 1
2
) (wn
i+1 − u)
+(1 + sgn(c)αn
i+ 1
2
)(wn
i − u)∣∣∣
≤ |c|(1 + αn
i+ 1
2
)maxk=0,1
∣∣wni+k − u
∣∣ .
QP
D D
1q q3wtsopi ,Zqo9so hw2au qckmq) AK A)% /1#z\+*J7 \ X7_#J ] /1b, c 0/1b)\Ö\_#X 01b& 7 \ Ni) αn
i+ 1
2
≥ 1 $ ∀(i, n) ∈ Z × N
ii) r|c|αn
i+ 1
2
+ αni− 1
2
2≤ 1 $ ∀(i, n) ∈ Z × N,
*J7+\^] 2&% ! ) %_! 7*J7'a! cbd]_ `'&7;#& JC 0 v# ±*J/^]6*
mini∈Z
wni ≤ min
i∈Z
wn+1i ≤ max
i∈Z
wn+1i ≤ max
i∈Z
wni .
mtXs'>tÔ¬E|Ì ÊA$ÐLÓMAE+ÅC=CIH?AÐE~'&FAGJ º . ® $Ì~@6GHD e
wn+1i = β0w
ni + β−1w
ni−1 + β1w
ni+1,
=β−1 =
rc
2
(1 + αn
i− 1
2
sgn(c))
=r|c|2
(sgn(c) + αn
i− 1
2
),
β0 = 1 − r|c|2
(αn
i+ 1
2
+ αni− 1
2
),
E~Dβ1 =
rc
2
(−1 + αn
i+ 1
2
sgn(c))
=r|c|2
(− sgn(c) + αn
i+ 1
2
).
,=C?)@®z?AEÖIHE|Ì ÊA$ÐLMAEÅ=CIJ?)ÐE~&FAGH Å$@|G&EÖIHE<)@6GJF)Ì$GJ<aEfMA? ÐL1ÇGJ?A ¿CGHIa|? ÓD(M L ÅC=CGH@
β−1 ≥ 0, β1 ≥ 0, β0 ≥ 0 E_D β−1 + β0 + β1 = 1.
Á Ez?AG¬E~|DfKL@6LFDG³<AL@IJE$(Ì~=CFAMAG,DGH=CFA (i) E~D (ii)º
g &k q3t su+1lnXopi B D F G Zi u´x3t CiAu αni+ 1
2
tu_k s t Q u~k^i k^t
hw2au qckmq) AK A A /1#z\+*J7 \ X7_#J ] /1b, c 0/1b)\Ö\_#X 01b& 7 \ Ni) ∃ γ ≥ 1 7_*5$#&7 ∀(i, n) ∈ Z × N, αn
i+ 1
2
= γ $
ii) r |c| γ ≤ 1 $ ∀(i, n) ∈ Z × N
7+\^] 2&% ! ) 7 \_: ´/1bd/1 /1bd7 O@
DDI D 1D
mtXs'>tÆF ?DGJIHGJ6LFDÑI =C<a$@6LUDE~?A@ H ¿>IJEV|Ì ÊA$ÐL-MAEÅC=IJ?AE$%AFAGJÀ Î $Ì$@|GHD+M)EvIÃL´ÐLFAGHÍ$@6E6?AG,ÅULFDE e
wn+1i = H
(wn
i−1, wni , w
ni+1
),
L ÅCE$ÌH(wn
i−1, wni , w
ni+1
)= β0w
ni + β−1w
ni−1 + β1w
ni+1,
= β0, β−1E_D β1
|=CFDÑM)!&FAGHM&LFA 0 º º ºË L<A@6Í~ (i) MAEfIÃL+<A@|=C<a=C6G,DGJ=F 0 º º _¿C=F´L αn
i+ 1
2
≥ 1 ¿ ∀(i, n) ∈ Z×N¿6?)<A<>=6=CFA®LIJ=@6
z? GHI³E_ÇGH|DE γ ≥ 1 ¿)D6E$I³z?AEª<a=C?A@D=?)D (i, n) ∈ Z × N αni+ 1
2
= γ ¿AMALFA(Ì$EªÌ$L β0, β−1E_D
β1M)E~ÅXGJE~FAFAE$FD
β−1 =rc
2(1 + γsgn(c)) =
r|c|2
(sgn(c) + γ) ,
β0 = 1 − r|c|γ,E~D
β1 =rc
2(−1 + γsgn(c)) =
r|c|2
(− sgn(c) + γ) .
Ô¬E|Ì ÊA$ÐL)À Î E$|Df=CFA=D6=CFAE+|G β0, β−1E_D β1
|=CFD<a=CDG3(=? Fz?AIH ºÁ =CÐE γ ≥ 1 ¿&LIH=C@6 β−1
E~D β16=CFDf<>=6GHD6G3 º
β1 ≥ 0 |G¬E$|Df|E$?AIHE$E$FDf6G r|c|γ ≤ 1, Ì~Ez?AG¬E~|DfKL@6LFDG³<&LU@fIJLÌ$=CF)MAGHD6GJ=CF (ii) ¿M =IHE+@6~6?AI,D LUD º
1 1 Q 5 462 4 $?$4DG¬?&9<C
=CGHE$FD f : R → R?)FAE 3=CF)Ì~DGH=CFMAE Ì~IÃL|6E C1 FA=CFWIJGJF) LGH@6E¿ w0 ∈ L∞(R) _D LFD IJLÌ$=CF)MAGHD6GJ=CF-GHFAGHD6GÃLIHEMA? <A@|=CBAIJÍ~ÐE º º
,=C?)@ γ ≥ 1 ¿ ÖFMA!AFAGHDfI E$F)6E$VBAIJE+|?AGHÅULFD e
X = w ∈ R/|w| ≤ γ||w0||L∞(R)E~Df=CF<a=C6E A = max
w∈X|f ′(w)| º
Ô¬E®6Ì Ê)$ÐL)À Î <a=C?A@IJE <A@6=BAIJÍ~ÐE FA=CFIHGJFA$LGJ@|EÊA=C=CKCÍ~FAE º _¿m $Ì~@6GHDyMAE®IJLÐLFAGHÍ$@6E6?AG,ÅULFDE e
wni+ 1
2
=1
2
(wn
i+1 + wni
)−
αni+ 1
2
2Sni+ 1
2
(f(wn
i+1) − f(wni )),
wn+1i = wn
i − r(f(wn
i+ 1
2
) − f(wni− 1
2
)).
º?>
OR
D D
Ô¬E6Ì ÊA~´L-MAEÅC=CIH?AÐE~%&FAGH 0 º?> E$DªÌ$=CFA|E$@ÅL1DG <&LU@ÑÌ$=FA|D6@6?AÌ_DGJ=FE~D+IJL3=CFAÌ_DGJ=F;MA??&?Ç Fz?A$@|GJz?AE+E$DÑM)=CFAFA~E<AL@
g(wni , w
ni+1) = f(wn
i+ 1
2
) = f(ϕ(wn
i , wni+1)),
L ÅCE$Ì eϕ(wn
i , wni+1) =
1
2
(wn
i+1 + wni
)−
αni+ 1
2
2Sni+ 1
2
(f(wn
i+1) − f(wni )).
,=C?)@ wni+1 = wn
i¿d=CFpL ϕ(wn
i , wni ) = wn
i;³GHI E$F)6?AG,Dz?AE g (wn
i , wni ) = f(wn
i ) ¿³E~D<&LU@6?AG,DEªIJE+|Ì ÊA$ÐLÓE$|DfÌ~=CFA6GH|DLFD º
1q q3wtsopi ,Zqo9so
ÖFFA=UDE a⊥b = min(a, b) E~D a>b = max(a, b)º
hw2au qckmq) AK A b \_#m''&/^\^7Ö5$#&7 f 7 \_ #bd7 /1bd]_ 0/1b&X7]_*,1\ \^7 C1 7_5$#h7 f ′ X7#b \_ bd7V] /1b)\_ b&0$ *J/1\+\^/1#z\+*J7 \ X7_#J ] /1b, c 0/1b)\Ö\_#X 01b& 7 \ N
1)Sn
i+ 1
2
|f ′(ani+ 1
2
)| ≤ αni+ 1
2
≤ γSn
i+ 1
2∣∣∣f ′(ani+ 1
2
)∣∣∣$ ∀(i, n) ∈ Z × N
$
2)r ≤ 1
γA$
7 ];N ani ∈
[wn
i+ 1
2
⊥wni− 1
2
, wni+ 1
2
>wni− 1
2
] 7_*Z5$#&7
f(wn
i+ 1
2
)− f
(wn
i− 1
2
)= f ′ (an
i )(wn
i+ 1
2
− wni− 1
2
)
7_an
i+ 1
2
∈[wn
i+1⊥wni , w
ni+1>wn
i
],
7_*5$#&7f(wn
i+1) − f(wni ) = f ′
(an
i+ 1
2
) (wn
i+1 − wni
),
*J7+\^] 2&% ! ) \$ J\ ~ c *J7('a_ cbd]_ `'&7+# JC 0 v# ±*J/^]6*
mini∈Z
wni ≤ min
i∈Z
wn+1i ≤ max
i∈Z
wn+1i ≤ max
i∈Z
wni .
mtXs'>tf E$DÖÌ~=CFDGJFz?AEVE~DÖM)$@6G,ÅULBAIJEª|?A@
[wn
i− 1
2
⊥wni+ 1
2
, wni− 1
2
>wni+ 1
2
] ;aIHED6ÊA$=C@|Í$EVMAE~ÖLÌ~Ì$@|=CGJ6E~ÐE~FzD6*AFAGJ~¿&FA=C?AMA=CF)FAEªI E_ÇGJDE$F)Ì$EªMAE an
i
º
.C
DDI D 1D
ËÑEf;~ÐE(|?A@ [wni ⊥wn
i+1, wni >wn
i+1
] ¿IHED6ÊA$=@6Í$EMAE$ LÌ~Ì$@6=GJ6|E$E$FD &FAGHFA=C?A MA=CF)FAEI E_ÇGJDE~FAÌ$EªMAE an
i+ 1
2
º I³ E~FA6?AG,DÑz?AE
wni+ 1
2
=1
2
(wn
i + wni+1
)−
αni+ 1
2
2Sni+ 1
2
f ′(an
i+ 1
2
) (wn
i+1 − wni
). º
ÆF<a=CLUFzD
δni+ 1
2
=αn
i+ 1
2
Sni+ 1
2
f ′(an
i+ 1
2
),
=CF =CB)DGHE$FDwn
i+ 1
2
=1
2
(1 + δn
i+ 1
2
)wn
i +1
2
(1 − δn
i+ 1
2
)wn
i+1.
Ö@fIJLÌ$=CF)MAGHD6GJ=CF (1) MAEªIÃL<)@6=C<a=C6G,DGH=CF 0 º º . ®MA=CF)FAE 1 ≤∣∣∣δn
i+ 1
2
∣∣∣ ≤ γ ¿AGJId E$FA|?AGHDfz?AE
|wni+ 1
2
| ≤ γ‖wn‖L∞(R) ≤ γ‖w0‖L∞(R) ≤ γ‖w0‖L∞(R).
Ë = wni+ 1
2
∈ X ∀(i, n) ∈ Z × NE~DL?A|6G an
i ∈ X E_DM)=CFAÌ f ′(ani ) ≤ A
ºÆFGHF1|E$Ì_D LFDfI $KLUIJGHD6 º ®M&LFAI ~DL<aE+Ì$=C@|@6E$Ì_DE~?A@=CF LU?A@L
wn+1i = H
(wn
i−1, wni , w
ni+1
)= β0w
ni + β1w
ni+1 + β−1w
ni−1,
º = L ÅCE$Ì e
β0 = 1 − r
2|f ′(an
i )|(∣∣∣δn
i+ 1
2
∣∣∣+∣∣∣δn
i− 1
2
∣∣∣),
β1 =r
2|f ′(an
i )| sgn (f ′(ani ))(∣∣∣δn
i+ 1
2
∣∣∣ sgn(f ′(an
i+ 1
2
))− 1)
=r
2|f ′(an
i )|(∣∣∣δn
i+ 1
2
∣∣∣− sgn (f ′)),
β−1 =r
2|f ′(an
i )| sgn (f ′(ani ))(∣∣∣δn
i− 1
2
∣∣∣ sgn(f ′(an
i− 1
2
))
+ 1)
=r
2|f ′(an
i )|(∣∣∣δn
i− 1
2
∣∣∣+ sgn (f ′)).
,=C?)@=CFD@|E$@+IHEv<A@6GHFAÌ$GH<>EM)?RÐL1ÇGJ?)¿aGJI |? ÓDMAEvD@6=?)ÅCE~@IHE$ªÌ$=FAMAGHD6GJ=CF)Ö6?A@ αni+ 1
2E~D r MAE+D6E$IJIHE+6=C@DE+z?AE β0¿ β1
E~D β−1|=CGJE~FD<>=C|GHD6G3E_D β0 + β−1 + β1 = 1
º Ö@=CF L
β1 ≥ 0 ⇐⇒ |δni+ 1
2
| ≥ sgn (f ′) , º P
β−1 ≥ 0 ⇐⇒∣∣∣δn
i− 1
2
∣∣∣ ≥ − sgn (f ′) , º @
M =³¿6=C?)ÐIJL9Ì~=CFAMAG,DGH=CF ∀(i, n) ∈ Z × N αni+ 1
2
≥Sn
i+ 1
2∣∣∣f ′(an
i+ 1
2
)∣∣∣¿=CFWL|6?A@|E β1 ≥ 0 ¿
β−1 ≥ 0
E~D β0 ≥ 0 ⇐⇒ r
2|f ′(an
i )|(∣∣∣δn
i+ 1
2
∣∣∣+∣∣∣δn
i− 1
2
∣∣∣)≤ 1.
Ö@f|=C?AIJE~(Ì$=CF)MAGHD6GJ=CFA(M)EIJL<A@6=<>=C|GHD6GJ=CF º º . ®=FLr
2|f ′(an
i )|(∣∣∣δn
i+ 1
2
∣∣∣+∣∣∣δn
i− 1
2
∣∣∣)≤ rAγ ≤ 1.
.)
D D
tXo oRt A AK + /1bd]_ 0/1b #4 #J b&#H ´%_! 5$#&7#;\^] 2&% 7 \_Ö] /1b& cb&#&7 `',"\^] 2X c $ 07_b&bd7 mtXs'>t,=C?)@D6=C?)D w ∈ X =CF L
∣∣g(wn
i , wni+1
)− f(w)
∣∣ =∣∣f(ϕ(wn
i , wni+1
))− f(w)
∣∣
≤ A∣∣ϕ(wn
i , wni+1
)− w
∣∣
≤ 1
2A
(1 +
αni+ 1
2
Sni+ 1
2
∣∣∣f ′(an
i+ 1
2
)∣∣∣)
maxj=0,1
∣∣wni+j − w
∣∣
≤ C maxj=0,1
∣∣wni+j − w
∣∣ ,
= C =A (γ + 1)
2
D tXopi 1r³sZtËÖLFAfIJE+Ì$L= e f ′
(an
i+ 1
2
)= 0 ¿&I ~D LU<>E+<A@|$MAGHÌ~D6E$?A@(MAE_ÅXGJE$FD
wni+ 1
2
=1
2
(wn
i + wni+1
).
g &k q3t
hw2au qckmq) AK A $ b\_#m''&/^\^75$#&7 f ′ X7®#bv\_ bd7] /1b)\_ b& /1#z\ *J7 \X7_#J] /1b, 0" 0/1b)\+\_#X 01b& 7 \;N
1) αni+ 1
2
= γSn
i+ 1
2∣∣∣f ′(ani+ 1
2
)∣∣∣$ ∀i ∈ Z
$ n ∈ N7 ] γ ∈ [1, γ] $ γ 7 \_(] /1b)\_0b& 7%$
2) rγA ≤ 1 7+\^] 2&% ! ) 7 \_: ´/1bd/1 /1bd7 mtXs'>tFAE Ì~=CFAMAG,DGH=CF |? LFDE <a=C?A@;@|E$|<>E~Ì~DE~@IÃL <)@6E$GJÍ~@6E Ì~=CFAMAG,DGH=CF MAE IJLW<A@|=C<a=C6G,DGJ=F0 º º . ®E~DfMAE)ÇE~@f?)F γ ∈ [1, γ] DE~I¬z?AE
αni+ 1
2
= γSn
i+ 1
2∣∣∣f ′(an
i+ 1
2
)∣∣∣. º R
.
DDI D 1D
ËÖLFAfÌ$EªÌ$LI ~D LU<>E+<A@|$MAGHÌ~D6E$?A@MAE_ÅGHE$FD e
wni+ 1
2
=1
2(1 + γ sgn (f ′))wn
i +1
2(1 − γ sgn (f ′))wn
i+1
E~DfI ~DL<aE+Ì$=C@|@6E$Ì_DE~?A@ ~Ì$@|GHD6=?AIJL93=C@6E+Ì$=FA6E~@|ÅULUDG,ÅCEª6?)GHÅULFDE
wn+1i = wn
i − r(f(wn
i+ 1
2
)− f
(wn
i− 1
2
)), 0 º ^
= H(wn
i−1, wni , w
ni+1
), 0 º C
L ÅCE$Ì H E~|DfI =C<a$@6LUDE~?A@(MA!AFAG¬M&LFA º = ºÔ¬EÑ6Ì Ê)$ÐL 0 º ^ E$D(Ð=FA=D=FAE|GdI =C<a$@6LUDE~?A@ H E$|DÌ$@6=GJ66LFD<&LU@®@LU<A<>=@|D+VÌ Ê&LÌ~?AFAEMAEªÌ$E~fÅUL@|GÃLBAIHE wn
i , wni−1
E_D wni+1
º
∂H∂wn
i−1
(wn
i−1, wni , w
ni+1
)=r
2(sgn (f ′) + γ) |f ′(an
i )| ,
∂H∂wn
i+1
(wn
i−1, wni , w
ni+1
)= −r
2(sgn (f ′) − γ) |f ′(an
i )|,
∂H∂wn
i
(wn
i−1, wni , w
ni+1
)= 1 − rγf ′(an
i ).
Ô¬ELUGHD(z?AE γ ≥ 1 E_D rγA ≤ 1 ¿AI =C<a$@L1DE$?)@ H E$DfÌ$@6=GJ66LFD<&L@@LU<A<>=@|D*vÌ Ê&LÌ~?AFAEMAEÌ$E~fÅUL@|GÃLBAIHE$~¿)GJId E$FA|?AGHDfz?AEIJE+6Ì Ê)$ÐL>)À* Î E~|DÐ=FA=D=FAE º
lZn#:7Xopt AK A /1 c w0 ∈ L∞ (R) $³\^/1#z\*J7 \Ñ] /1b, c 0/1b)\-X% b& 07 \-b)\*,('a / '&/^\_ c 0/1b! ) ) *, \^/1*.#X 0/1bWb&# ´%_! 5$#h7LX/1b&bd% 7+')´*J7Ó\^] 2&% ] /1b 7_ X7)7_\Ð* #b& 5$#&7\^/1*.# 0/1bp7_b& / 'a 5$#&7 mtXs'>tÔ¬E |Ì ÊA$ÐL 0 º > dE$Dy?)F|Ì ÊA$ÐLD6@6=CGH¬<>=CGHFD¬Ì$=FA6E~@|ÅULUDG8AÌ~=CFA6GH|DLFD=CFA=UD=CFAE `ϪÀ R¿6=C?)(IJE$®Ì$=CFAM)GHDGH=CFA MAE+IÃL<A@6=<>=C|GHD6GJ=CF 0 º º _¿XGJIa E$FA|?AGHDz?AE+Ì~EÖ6Ì Ê)$ÐLÓE~|D(Ì~=CFA|GJ|DLFDL ÅCE$̪D=?)DEªÌ$=FAMAGHD6GJ=CF-M E~FzD6@6=C<)GJE º
1 1 S CÖ 9A E7:9<CJADCJ=24 7:9$C AÑ4D5 _T G7)C69<GKAD?$=7CK;9
ÔZLÌ~=CFAMAG,DGH=CF;MAEÓ|DLBAGHIJGHD6 (2) MAEvIÃL <A@|=C<a=C6G,DGJ=F 0 º º . fE_ÇGJKCEvz?AE |f ′(ani )| |=CGHD%&FAG º
FAEªÌ~=CFAMAG,DGJ=F-6? LUFzD6EV<a=C?A@(Ì~E$IÃL¿AE$Dz?AE aniE~D wn
i+ 1
2
@6E~|D6E$FDÑBa=C@|FA$ º
. .
D D
ËÖLFAfÌ$Eªz?AG¬<A@|$Ì~Í$MAEU¿aÌ~E$IJLLv_Dª=CB)D6E$Fz?³¿&E$F GJ<a=CLFD(z?AE wni+ 1
2
∈ X ¿A= X E~|DMA!&F)GL? M)$BA?)DÑMAEVÌ~E~D|DEV|E$Ì_DGJ=FÅXGÃLÓ?AF;<&LU@LÍ~D6@6E γ ≥ 1
º f=C?AÅC=C?AIH=CFAGJÌ$G¬IHE~ÅE$@ÑI GHFAMA_DE$@ÐGHF&LUD6GJ=CFGHFzD6@6=XMA?AG,DEv<&L@ªÌ~EÐ<&LU@LÍ~D6@6E γ º ,=C?)@Ì~E$IÃL¿¬=CFp@|E$ÐL@6z?AEÓz? ?AFAE´Ì~=CFAMAG,DGH=CF6? 6LFDEV<a=C?A@(z?AE wn
i+ 1
2
@6E~|D6EBa=C@6F)+E$|Dfz?AE wni+ 1
2
∈[wn
i ⊥wni+1, w
ni >wn
i+1
]. Ö@
wni+ 1
2
=1
2
(1 + δn
i+ 1
2
)wn
i +1
2
(1 − δn
i+ 1
2
)wn
i+1,
?AFAEªÌ$=FAMAGHD6GJ=CF-|? LFD6EVE$DfMA=CFAÌ 1 + δni+ 1
2
≥ 0 E_D 1 − δni+ 1
2
≥ 0 ¿AGJI³ E~FA6?)GHDfz?AE
∣∣∣δni+ 1
2
∣∣∣ ≤ 1 ⇐⇒ αni+ 1
2
≤Sn
i+ 1
2∣∣∣f ′(an
i+ 1
2
)∣∣∣.
Á E$IJLL ÅCE~ÌÑIJLªÌ~=CFAMAG,DGH=CF (1) MAEIJLª<)@6=C<a=C6G,DGH=CF 0 º º . yMA?´<)@6GJF)Ì$GJ<aE(MA?дL1ÇGH?A GHÐ<a=C|Ez?AE αn
i+ 1
2
=Sn
i+ 1
2∣∣∣f ′(an
i+ 1
2
)∣∣∣º¬» GHF&LIHE$E$FD+=CF9MA~MA?AGHDIHEv6Ì ÊA~´L )À* Î 3Ì E$|DIJEv|Ì ÊA$ÐL
)À* Î E~F LGJ6LFD®GHFzD6E$@ÅCE$F)GJ@IHEÑ6GHKCFAEMAE f ′ M&LFAI _D L<aEÑ<)@6$M)GJÌ~D6E$?A@ z?AGd ~Ì$@|GHD|=C?A(IJL3=C@6E
wni+ 1
2
=1
2
(wn
i + wni+1
)− 1
2∣∣∣f ′(an
i+ 1
2
)∣∣∣
(f(wn
i+1) − f(wni ))
wn+1i = wn
i − r(f(wn
i+ 1
2
)− f
(wn
i− 1
2
)),
6=CG,D
wni+ 1
2
=1
2
(wn
i + wni+1
)− 1
2sgn
(f ′(an
i+ 1
2
)) (wn
i+1 − wni
)
wn+1i = wn
i − r(f(wn
i+ 1
2
)− f
(wn
i− 1
2
)).
D tXopi 1r³sZtG =CF 6?)<A<>=6E-z?AE f ′ KL@|MAE´?)F 6GJKFAE´Ì~=CFA|DLFDE_Dvz?AE I =CFR$Ì$@|GHDIJEÐ6Ì ÊA~´L À Î 6=C?)IJL93=C@6E
wn+1i = H
(wn
i−1, wni , w
ni+1
),
I E$@|@6E$?)@fM)EªD6@6=CF)Ì LUD6?A@6E+E~|DMA=CFAFA~Eª<&L@ #=CGH@(ÀfL^ÅXGJL@|D(E_DÑϪ=XMAIJE 6ÉzGRQR `ϪÀ R
ET (w)(x, t) = −τ ∂∂x
[B (w, r)
∂w
∂x
]+ (τ 2)
L ÅCE$Ì B (w, r) =1
2r2
j=1∑
j=−1
j2 ∂H∂wn
i+j
(w,w, w)− 1
2f ′(w)
2 ºÁ =CÐE
∂H
∂wni+1
(w,w, w) =r
2(1 − sgn (f ′)) (f ′ (w))
E~D∂H
∂wni−1
(w,w, w) =r
2(1 + sgn (f ′)) (f ′ (w)) ,
. >
DDI D 1D
LIJ=@6
B (w, r) =1
2r2r |f ′(w)| − 1
2f ′(w)2
=|f ′(w)|
2r(1 − r |f ′(w)|) .
Ô¬EÖ6Ì ÊA~´LE~|D(M)=CFAÌÖz?&L|GdM =C@|MA@6E 2 6G>IÃLÌ~=CFAMAG,DGH=CF´MAE|DLBAGJIHGHD6E~|D(MAEÑI =C@6MA@|EM)E 1º G
IÃLvÌ$=FAMAGHD6GJ=CF-M)ED LB)GJIJG,DE$|DfGHF3$@|GJE$?)@6E 1 IJE+6Ì Ê)$ÐLE~|DfM =C@6MA@|E 1º
1 1 GK5@; 7) 7@5 4 ; EHG9? ;DCK5
r³sZi)kmq t9ht#UuÔ ~X?ALUDGH=CFÐM)EÑÄ)@6KCE~@6 @6E$<)@6$|E$FDE?AFAEfIJ=CG)MAEÌ~=CFA|E$@|ÅULUD6GJ=CFFA=CFIJGHFA LUGJ@6E®Ì$=CB)GJFA~EL ÅE$Ì?AFAE+<&LU@|DGHEÖFA=CF-Ê)=CÐ=KCÍ$FAEÑMAG >?A6G,ÅCEU¿)=MA~IJGHLFDM)EÖ´LUFAGJÍ~@6EÖ|GJ<AIJG8&$Ef?AF-$Ì~=C?AIHE$E$FDE~Dfz?AG¬ $Ì~@6G,D e
∂w
∂t+
∂
∂x(w2
2) = ν
∂2w
∂x2,
= w E$D?AFAEÖÅUL@|GÃLBAIHEÑÌ~=CFA6E~@|ÅULUD6GHÅCEU¿ ν E$Df?AFÌ$=XE Ì$GHE$FDMAEÖÅXGJ|Ì$=C|GHD º ÖFÌ$=CF)6GJM)Í$@6E+IHEÌ L= ν DE$F)M ÅE$@6 0º ËÖLFAfÌ$EªÌ LUfI ~z?&LUDGH=CF-MAE_ÅGHE$FD e
∂w
∂t+∂(w2
2)
∂x= 0,
E~Df=CF<A@|E$FAMÌ$=CÐEÑÌ$=CFAM)GHDGH=CF-GJF)GHDGJLIJE
w0(x) =
0 |G 0 ≤ x < 0.2,
1 |G 0.2 ≤ x < 0.4,
1 − 5(x− 0.4) |G 0.4 ≤ x < 0.6,
0 6GHFA=CF .Á% E~|D?AF <)@6=CBAIHÍ$EMAE ÀfGHE$ÐLFAFDE~|D6R<&LU@ » º ËÑE#?)ÈX|D Ë #+ > 0¿(MA=FzD IJL |=CIJ?DGJ=FÌ$=CFD6GJE$FD+?)FAE=CFAM)EMAE@6L@6 LÌ~D6GJ=CF;z?AG6E93=@6E ´<&LU@|DGH@ÑMAEVIÃLÐ<)@6E$GJÍ~@6EMAGJ|Ì$=CFDGHFz?AGHD6E~D´X?)GÑÌ$=Ð<A@|E$|6E;IJL ~=CFAE;E~FzD6@6E x = 0.4 E_D x = 0.6 ¿ E_D ?AF)E =CFAM)E;MAE Ì Ê)=ÌX?)GÑ6E<A@6=<&LKCE*+<&L@DGJ@yMAE x = 0.6 L ÅCE~Ì?)FAEfÅXG,DE$|6E$KCLIJE2 σ =
1
2
º ÖFÐÌ~=CÐ<AL@6E(IHE$@6~6?AI,D LUD6Fz?AÐ~@6GHX?)E$MA=CFAF)$<&L@IHE(6Ì ÊA~´L")À* Î E_D IHE6Ì Ê)$ÐL Î ÈB)@6GJM)EÔLmÇpE~FAMA@|= − ÔZL1Ç7» @6GJE~MA@6GHÌ ÊAÌ$=CFA½~?v<&L@ »®º ËÑE+#?)ÈX|D Ë #Ö > )E_D IJLf|=CIJ?)D6GJ=CFDÊA~=C@6GHz?AEL?XÇV<;$E$GJF)|D LUFzD6t = 0.1s E_D t = 0.6s
º ÔZL Á» ÔpÅUL?)D 1 <a=C?A@fIJE6Ì ÊA~´LEÀ Î E_Dª º <a=C?A@ÑIHE|Ì ÊA$ÐLÊÈB)@6GJM)E;E~D´=CF=CFD@6EIJE~?A@´Ì~=C?A@6BaE M E~@6@6E~?A@EpI GHFA|DLFDÐD+ º =U º ,?)GJ6z?AE IJE |Ì ÊA$ÐL
.
D D
)À* Î ´M)E~ÅXGJE~FzDIHEÑ|Ì ÊA$ÐLvMAE+ÔZL1Ç7 RE$FAM)@6= <>=C?)@IJE<&L@LUÐÍ_D@6EMAE+Ì$=FzD6@VSCIHE αni+ 1
2
= 1E~DfIJEª|Ì ÊA$ÐLÐÊÈXBA@6GHMAEª$KLUIJE$E$FD<a=C?A@IJEª<&LU@LÍ~D6@6E θn
i+ 1
2
= 0 ¿&FA=C?)LUIJIJ=FAL Ì ÊAE$@fIJE~Ì$=C?)@6BaE$fMAEª@6$<AL@|D6GHDGH=CF MAE αn
i+ 1
2
E~D θni+ 1
2
ºËÑEª<AIJ?)IJE~<&L@6LÐÍ_D@|E$ αn
i+ 1
2
E~D θni+ 1
2
Ì$=FzD6@VSCIHE$FD(IJLMAG >?A6GH=CF Fz?A$@|GJz?AE¿)=CF-@6E~´L@|z?AEz?AEVÌ~E$IJIHEªÌ$GE$|D$IHE~Å$EV<a=C?A@fIJEª|Ì ÊA$ÐLE)À* Î ÐI E~FAMA@|=CGHDM ?AFAE=CFAMAEMAE@L@|!LÌ_DGJ=FE~DÑL?A|6GGJ<>=C@D LFD6EÖ<>=?A@IHEª6Ì ÊA~´LÊzÈXBA@|GJMAE"ÓI GJ6|?AEMA? <A@|=C<&LKCLUDGH=CF-M ?AFAEV=FAMAEMAEÌ ÊA=XÌ ºÔ¬E?A?ÇFz?A$@6GHz?AEªMA?|Ì ÊA$ÐLÓÊzÈXBA@|GJMAEª ~Ì$@6G,DfMAEªIÃL9L½~=CF|?AGHÅULFD6E
φni+ 1
2
(θn
i+ 1
2
;W ni ,W
ni+1, r
)= θn
i+ 1
2
φMLFi+ 1
2
(W n
i ,Wni+1, r
)+(1 − θn
i+ 1
2
)φLW,ε
i+ 1
2
(W n
i ,Wni+1, r
),
= φMLF (U, V, r) E~DªIHE9?&?Ç;Fz?AÐ~@6GHX?)EvMA?96Ì ÊA~´L-MAEÓÔZL1Ç7 » @|GJE$M)@6GJÌ Ê)ÖÐ=XMAG8&VMA=FAFA<&L@φMLF (U, V, r) =
1
2(f(U) + f(V )) − 1
4r(V − U)
E~D φLW,ε (U, V, r) E~DI LU<A<A@6= ÇGJÐLUD6GJ=CFMA? ?&?ÇFz?AÐ~@6GHX?)E6?AG,ÅULFD?AFAEM)$@6G,ÅC~E(MAGJ@|E$Ì~D6GJ=CFFAE$IHIJE+E_D?)F <AL@LÍ~D6@6E ε MA? 6Ì ÊA~´LÓMAEÔZL1Ç7 RE$FAMA@|= 9MA=CF)FAª<&L@
φLW,ε (U, V, r) =1
2(f(U) + f(V )) − r
2
f [ϕ (U, V ) + ε (f(V ) − f(U))] − f (ϕ (U, V ))
ε,
= ϕ (U, V ) E$|Dª?)F;~DLUD+=mÈE$F6LUDGH LGHLFDIÃLÐ@|E$IJLUDGH=CF MAEVÌ$=CFA|GJD LFAÌ~E ϕ (U, U) = U ¿E$D®Ì ÊA=GJ6G&D6E$Ihz?AEIJLÐLUD6@6GJÌ~E A (U, V ) = A (ϕ (U, V )) ¿E~D IJE'?&?ÇFz?A$@|GJz?AEfMA?´|Ì ÊA$ÐLMAEÔZL1Ç7 RE$FAMA@|= 9E$DfMA=CFAFAª<&LU@
φLW (U, V, r) =1
2(f(U) + f(V )) − r
2A (U, V ) (f(V ) − f(U)) .
θni+ 1
2
E$DÌ ÊA=GJ6GaMAEfDE$IHIJE6=@|DEz?AEÖIJEDE~@6EM)EÑIÃLMAGJ|6GH<&LUDGH=CF ηni+ 1
2
|=CGHD GJF53$@6GHE$?A@=C?´$KLI 0 E_D
ηni+ 1
2
= S(Un+1i ) − S(Un
i ) + rDUS
ni− 1
2
(Un
i − φni− 1
2
)+DUS
ni+ 1
2
(φn
i+ 1
2
− Uni
),
= S(U) =1
2‖U‖2 E~|DfIÃLvFA=@6E+MAEªI $FAE~@6KCGHE¿AE_D DUS
ni+ 1
2
= DUS
(Un
i + Uni+1
2
) º
.Q=
DDI D 1D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
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0.2
0.3
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2
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2
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i+ 1
2
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0.2
0.3
0.4
0.5
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0.8
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2
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−11 −10 −9 −8 −7 −6 −5 −4−10
−9
−8
−7
−6
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Log(dx)
Lo
g(L
1−
err
eu
r)
Representation des courbes d’erreur de SRNHS et Hybride
hybrideSRNHS
º P +,-.0/ -1243516798:9;9<=8> ?2435-67@3BAC D E #"$ & -FG,1 ' 0.734
.Q@
DDI D 1D
+* © ¤ ÂZ- £_¤ -¤ ³² ³©Zµ ¤-¢ ¢¯µ-¤
11 Q 5 $?$4DG¬?&9<C
ÖFÌ~=CFA6GHMAÍ$@|EªIJE+|ÈXDÍ$EÊzÈX<aE$@|B>=CIHGJz?AE+IHGJFA$LGJ@|EÑÊA=C=CKCÍ~FAEÓ^Ë 6?AG,ÅULFD e
∂W
∂t+ A∂W
∂x= 0, (x, t) ∈ D × R
∗+, D ⊂ R
W (x, 0) = W0(x), x ∈ D0 º m
L ÅCE$Ì W : D × R+ → Ω ?)F ÅCE$Ì_DE~?A@ m Ì$=C<a=CLFD6E$MAE´X?ALFDGHD6$Ì~=CFA|E$@|Å$E~$¿=ÅUL@6GJLBAIJE~vM ~D L1D ¿ Ω _D LFDÐ?AF=C?)ÅCE~@|DÐB>=@6FAMAE
Rm ¿ A E$D´?AF)EÐLUD@|GJÌ~E Ì L@|@6~EM&LUFA
Mm(R)º
,=C?)@VÌ$E´<A@6=CB)IJÍ$EIJE6Ì ÊA~´L À Î <A@|$6E~FD-<&L@ »®º Ä®E$F)ÉXÊALIJMA=?AF M&LF) Ä®E$F&C®E_DM&LFAIHE+<A@6E~ÐGHE$@(Ì Ê&L<)GHD@|E+MAEªÌ$E_D6DEªDÊ)Í$6E $Ì~@6G,DMAEªIÃLv´LUFAGJÍ~@6EÖ|?AGHÅULFDE e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
A(W n
i+1 −W ni
)
W n+1i = W n
i − rA(W n
i+ 1
2
−W ni− 1
2
)
=Sn
i+ 1
2
= maxp=1,...,m
(|λp|) = ρ (A) ,
L ÅCE$Ì ρ(A) IJE@L ÈC=F;6<aE$Ì_D@LUIMAE A ¿ [λp]p=1,...,m|=CFDÖIJE~fÅULIHE$?A@|<)@6=C<A@|E$ÖM)EVIÃLÓÐLUD@|GJÌ$E
A E_D αni+ 1
2
E$Df?AF<AL@LÍ~D6@6E+MAEªÌ~=CFD@VSIJE ºÔ¬Ev|ÈX|D6Í$E º ^ E$DÊzÈX<aE$@|B>=CIHGJz?AEU¿³GJI E$FA|?AGHDªz?AEÓIÃL ÐLUD6@6GJÌ~E A E$DM)GÃLKC=F&LIJGHLB)IJEM&LFA(?)FAE+B&L|E+MAE$(ÅCE~Ì~D6E$?A@|f<)@6=C<A@|E$ º +F-F)=DE λ1 ≤ λ2 ≤ ... ≤ λm
6E~ÅULIHE$?A@|<A@|=C<A@6E~L6|=Ì~GJ~E$ L?ÇÓÅCE~Ì~D6E$?A@|®<A@|=C<A@6E~ r1, ..., rm ¿zE~D =CFFA=D6E B = r1, ..., rm IÃL+B&L|EMAE R
m
3=C@6$Eª<&LU@fIHE$(ÅCE~Ì~D6E$?A@|Ñ<A@|=C<A@6E~M)E A º X=CGHD R = [r1, ..., rm] IÃLvÐLUD@|GJÌ$E+M)E<AL66LKCEMAEIÃL-B&L|EÐÌ$LFA=CF)GJz?AEÓMAER
m IÃL BAL6E B ¿¬GJI E~FA6?AG,DVz?AEIÃL-ÐLUD@|GJÌ~E A = RΛR−1 ¿¬=Λ E~|DÖIJLÐÐLUD@|GJÌ~EMAE~ÅULIHE$?A@|Ñ<A@|=C<A@6E~ÖE~D R−1 E~|D+IJLÐÐLUD@|GJÌ$EªGHFzÅE$@|6EMAEVIÃLÓ´LUD6@6GHÌ$EMAE<&L|LKCE ºÔ _D L<aEª<A@6~MAGJÌ_DE$?)@f ~Ì$@|GHD
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
A(W n
i+1 −W ni
).
ÖF<a=C6E V = R−1W ¿&I _D L<aEÖ<A@|$MAGHÌ~D6E$?A@MAE_ÅGHE$FD
V ni+ 1
2
=1
2
(V n
i+1 + V ni
)−
αni+ 1
2
2Sni+ 1
2
Λ(V n
i+1 − V ni
)
.QR
: D D
E~DfI ~DL<aE+Ì$=C@|@6E$Ì_DE~?A@M)E~ÅXGJE~FzD
V n+1i = V n
i − rΛ(V n
i+ 1
2
− V ni− 1
2
).
Ô ~Ì$@6G,D?A@|EªE$|DfMA=FA̪$z?AGHÅULIHE$FDEL?|Ì ÊA$ÐLM)EªÅ=CIJ?)ÐE~2AFAGJ*)À* Î |?AGHÅULFD e
V ni+ 1
2
=1
2
(V n
i+1 + V ni
)−
αni+ 1
2
2Sni+ 1
2
Λ(V n
i+1 − V ni
),
V n+1i = V n
i − rΛ(V n
i+ 1
2
− V ni− 1
2
),
L<A<AIHGJz?AªL?ÈDÍ~ÐE e
∂V
∂t+ Λ
∂V
∂x= 0, (x, t) ∈ D × R
∗+, D ⊂ R,
V (x, 0) = V0(x), x ∈ D.Á Ev|ÈX|D6Í$EÓE$|Dª~z?AGHÅULIHE$FD m $z?&LUD6GJ=CFAÑGJFAMA~<>E~FAM&LFD6E$ m ÅUL@|GÃLB)IJE$~¿>Ì E~|D MAGJ@|Ez?AEIHE+<A@6=BAIJÍ~ÐEÖ|E@6LÍ$FAE e ∀p ∈ 1, ..., m e
∂vp
∂t+ λp
∂vp
∂x= 0, (x, t) ∈ D × R
∗+, D ⊂ R,
vp(x, 0) = (vp)0(x), x ∈ D,E~DfIJE+|Ì ÊA$ÐL)À Î $Ì$@|GHDf<a=C?A@(Ì$E~<A@6=CB)IJÍ$E$ 1D |Ì LIJLGJ@|E¿ ∀p
(vp)n
i+ 1
2
=1
2
((vp)
n
i+1 + (vp)n
i
)−
αni+ 1
2
2Sni+ 1
2
λp
((vp)
n
i+1 − (vp)n
i
),
(vp)n+1i
= (vp)n
i− rλp
((vp)
n
i+ 1
2
− (vp)n
i− 1
2
).
hw2au qckmq) AK )% /1#z\+*J7 \ X7_#J ] /1b, c 0/1b)\Ö\_#X 01b& 7 \ N
1) αni+ 1
2
≥ ρ(A)
|λp|∀(i, n) ∈ Z × N
$ ∀p ∈ 1, ..., m 7_*Z5$#&7 λp 6= 0 $
2) r ≤ ρ(A)−1 7+\^] 2&% Ó\$ J\ ~ c*J7('a! cbd]_ `'&7;#A JC 0 v# ±*J/^]6*
mini∈Z
(vp)n
i≤ min
i∈Z
(vp)n+1i
≤ maxi∈Z
(vp)n+1i
≤ maxi∈Z
(vp)n
i.
mtXs'>tÖF L ∀p
(vp)n+1i
= (vp)n
i− r
2λp
[((vp)
n
i+1 + (vp)n
i
)−αn
i+ 1
2
Sni+ 1
2
λp
((vp)
n
i+1 − (vp)n
i
)]
+r
2λp
[(vp)
n
i+ (vp)
n
i−1 −αn
i− 1
2
Sni− 1
2
λp
((vp)
n
i− (vp)
n
i−1
)].
>
DDI D 1D
ÆF L<)<AIJGHX?ALFDIJE+<;$E+@LGH6=CFAF)E$E$FDz?AEªM&LFAIJE+Ì$L6Ì$LIÃLGH@6EU¿A=CF$Ì~@6GHD
(vp)n+1i
= βni−1(vp)
n
i−1 + βni (vp)
n
i+ βn
i+1(vp)n
i+1,
L ÅCE$Ìβn
i−1 =r
2λp
(1 +
αni− 1
2
Sni− 1
2
λp
),
βni = 1 − r
2
(αn
i− 1
2
Sni− 1
2
+αn
i+ 1
2
Sni+ 1
2
)λp
2,
βni+1 =
r
2λp
(αn
i+ 1
2
Sni+ 1
2
λp − 1
).
ÖF<a=C6E [δni+ 1
2
]p
=αn
i+ 1
2
Sni+ 1
2
λp
º
1) ÆFRL<A<AIHGJz?&LFDªIJEv<;$EÓL@|KC?AE$FDªz?AEÓM&LFAªIJEvÌ$Lª6Ì$LIÃLGH@6EU¿d=CFRL?A@6L βni+1 ≥ 0 E~D
βni−1 ≥ 0 |=C?AIÃLvÌ~=CFAMAG,DGJ=F-6? LUFzD6E αn
i+ 1
2
≥Sn
i+ 1
2
|λp|¿ ∀(i, n) ∈ Z × N
¿ ∀p ∈ 1, ..., m º
2) Ë LU<A@6Í~ 1) ¿>=CF MA$MA?)GHDÖz?AE 1αn
i+12
Sn
i+12
+αn
i− 12
Sn
i− 12
≤ |λp|2
∀p ∈ 1, . . . , m ;dE_DÖIÃLÌ$=CF)MAGHD6GJ=CF
βni ≥ 0 E$DLUD6GJ LGHD6EÑ6G rλp
2 ≤ 2(αn
i+12
Sn
i+12
+αn
i− 12
Sn
i− 12
) ∀(i, n) ∈ Z ×N¿ ∀p ∈ 1, ..., m ¿)Ì~E
z?AG¬E$DÅE$@6G8&ª6=C?)fIJLvÌ$=CFAM)GHDGH=CF-6? Ð6LFDE rρ(A) ≤ 1º
11 9A E7:9<CJADC = 24 7:9$C 5@2;D5 2 9<EHC E 7@9?$=?$C¬ +C
ËÑE ÐLFAGHÍ$@6E´LF&LIH=CKC?AE I _D?AMAE<LGHD6E´M&LUFAIJEÌ LU6Ì$LIÃLGH@6EU¿=CFp6E <A@|=C<a=C6E M)E´Ba=C@6FAE~@I ~DLUD GJFDE~@6$MAGJLGJ@|E (vp)
ni+ 1
2
∀p ∈ 1, . . . , m º ,=C?)@®Ì$E~IÃLÌ ÊALz?AEÖGJF)|D LUFzD =CFÐGJ<>=6E6?A@ vn
i+ 1
2
IÃLvÌ$=CF)MAGHD6GJ=CF-|? LFDE6?)GHÅULFDE e
(vp)n
i+ 1
2
∈[(vp)
n
i+1⊥(vp)n
i, (vp)
n
i+1>(vp)n
i
]∀ (i, n) ∈ Z × N
E~D ∀p ∈ 1, ..., m.0 º /.
Ö@(vp)
n
i+ 1
2
=1
2
(1 + [δn
i+ 1
2
]p
)(vp)
n
i+
1
2
(1 − [δn
i+ 1
2
]p
)(vp)
n
i+1,
<>=?A@(z?AEIJLÌ$=CF)MAGHD6GJ=CF º . ®6=CG,D@6E~Ð<AIHGJEU¿)GJI>6? ÓDM L^Å=CGJ@
1 +[δni+ 1
2
]p≥ 0 E~D 1 −
[δni+ 1
2
]p≥ 0,
>
: D D
GJI³ E~FA6?)GHDfz?AE∣∣∣[δn
i+ 1
2
]p
∣∣∣ ≤ 1 ∀p ∈ 1, . . . , m, ∀(i, n) ∈ Z × N.
» GJFALIJE~ÐE~FzD6=C?AIJLÓÌ~=CFAMAG,DGH=CF 0 º /. !¿)=CFD@|=C?)ÅE
∀p ∈ 1, . . . , m∣∣∣[δn
i+ 1
2
]p
∣∣∣ ≤ 1. º >
Ô¬E$+MAE~?Ç9GJF)$KLIHGHD6$ αni+ 1
2
≥Sn
i+ 1
2
|λp|3Ì$=CFAM)GHDGH=CF;MAEvD LB)GJIJG,DMAEvIÃLÐ<A@6=C<a=C|GHDGH=CF º . º E_D
αni+ 1
2
≤Sn
i+ 1
2
|λp|GJ<AIJGHz?AE$FD ∀(i, n) ∈ Z × N
E_D ∀p ∈ 1, ..., m αni+ 1
2
=Sn
i+ 1
2
|λp|º ÖFÅC=CG,D
MA=CFAÌz?AEIHE<AL@LÍ~D6@6EMAEfÌ$=CFD6@VSCIHE αni+ 1
2
M)$<aE$FAM M)E$ ÅULIHE$?A@| <)@6=C<A@|E$~¿M =ÐIJEÌ ÊA=CG ÇMAEIJE+<A@|E$FAM)@6E6=C?AIJL3=@6EÖM ?AFAE´LUD6@6GHÌ$EÖMAGJLKC=CFALIJE
αni+ 1
2
=
Sn
i+12
|λ1| 0 . . . 0
0Sn
i+12
|λ2| . . .
ºººººº
0
º º º0
0 . . . 0Sn
i+12
|λm|
.
Ö@ V = R−1Wº ÆF@|E$<AIÃL½$LFDM&LFAI _D L<aE+<A@6~MAGJÌ_DE~?A@$¿&=CF=CBDGJE~FD e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2Sni+ 1
2
Rαni+ 1
2
ΛR−1(W n
i+1 −W ni
)
=1
2
(W n
i+1 +W ni
)− 1
2Sni+ 1
2
Rαni+ 1
2
R−1A(W n
i+1 −W ni
).
ÖF<a=C6EQn
i+ 1
2
= Rαni+ 1
2
R−1.» GJFALIJE~ÐE~FzDIJE+6Ì Ê)$ÐL>)À* Î MAE_ÅXGJE$FD
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2Sni+ 1
2
Qni+ 1
2
A(W n
i+1 −W ni
),
W n+1i = W n
i − rA(W n
i+ 1
2
−W ni− 1
2
).
ÆF|GJ<AIJG8hLFD=CF =CB)D6GJE~FzDIHE+6Ì ÊA~´L)À* Î
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2sgn (A)
(W n
i+1 −W ni
),
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
)),
>
DDI D 1D
L ÅCE$Ì sgn (A) = R sgn (Λ)R−1 E~D sgn (Λ) = diag
(λk
|λk|
) ºD tXopi 1r³sZt,=C?)@?AF <A@|=CBAIJÍ~ÐE+IHGJFA$LGJ@|E F (W ) = AW IHEª6Ì ÊA~´L)À* Î 6E@LÍ$F)E"v?AF6Ì ÊA~´LMA$Ì~E$FD@|L=CFD(Ì~=CÐE+IJE+|Ì ÊA$ÐLÓMAEÀf=XE¿# » »Á E_D*# » À=EÖE~FE dE_D e
ΦV FFC(W n
i ,Wni+1
)=
1
2A(W n
i +W ni+1
)− 1
2sgn (A)A
(W n
i+1 −W ni
)
=1
2A(W n
i +W ni+1
)− 1
2|A|(W n
i+1 −W ni
)
= ΦRoe(W n
i ,Wni+1
).
ΦSRNHS(W n
i ,Wni+1
)=
1
2A((W n
i +W ni+1
)− sgn (A)
(W n
i+1 −W ni
))
= ΦV FFC(W n
i ,Wni+1
).
Ë L?)D@|E<&L@D-6GÖ=F<)@6=1|E_D6DE W ni+1 − W n
iMALFA IJL B&L6EM)E$ ÅCE~Ì~D6E$?A@|<A@|=C<A@|E$-MAEIJL
´LUFAGJÍ~@6E+6?)GHÅULFDE W ni+1 −W n
i =
p=m∑
p=1
γprp=CFL?A@L
W ni+ 1
2
= W ni +
1
2
(W n
i+1 −W ni
)− 1
2sgn (A)
(W n
i+1 −W ni
)
= W ni +
1
2
p=m∑
p=1
γp (1 − sgn(λp)) rp
= W ni +
∑
λp<0
γprp
= WV FRoe
(0,W n
i ,Wni+1
),
= WV FRoe
(0,W n
i ,Wni+1
) E~|DZIJL6=CIH?)DGH=CFÖMA?ª<A@|=CBAIJÍ~ÐEMAE ÀfGHE$ÐLFAF³¿ γp = tlp·(W n
i+1 −W ni
)E~D tlp
<>=?A@ p = 1, . . . , m IHE$ÖÅCE~Ì~D6E$?A@|V<A@|=C<A@6E~9-KLU?AÌ ÊAEMAE A ÅC~@6G8hLFD tlp · rq = δpqM =<a=C?A@(Ì~EªÌ LfIJGHFA LGH@6EΦSRNHS
(W n
i ,Wni+1
)= AW n
i+ 1
2
= AWV FRoe
(0,W n
i ,Wni+1
)= ΦV FRoe
(W n
i ,Wni+1
).
11 Q 5 462 4 $?$4DG¬?&9<C
ÖFÌ~=CFA6GHMAÍ$@|EªIJE+|ÈXDÍ$EÊzÈX<aE$@|B>=CIHGJz?AE+FA=FIHGJFA$LGJ@|EÑÊA=C=CKCÍ~FAEv^Ëj6?AG,ÅULFD e
∂W
∂t+∂F (W )
∂x= 0, (x, t) ∈ D × R
∗+, D ⊂ R
W (x, 0) = W0(x), x ∈ D0 º m
> .
: D D
L ÅCE$Ì W : D × R+ → Ω ?)F ÅCE$Ì_DE~?A@ m Ì$=C<a=CLFD6E$MAE´X?ALFDGHD6$Ì~=CFA|E$@|Å$E~$¿=ÅUL@6GJLBAIJE~®M ~DLUD ¿XÌ$=ÐEÑIJLV´LU66EU¿)IJLz?&LFDG,DÖM)EÖÐ=?)ÅCE~ÐE~FzDE~D(I ~FAE$@|KCGJEM&LFAIHEÑÌ$LMAE$<A@6=BAIJÍ~ÐE~MAE IÃLÐ~Ì LF)GJz?AE MAE$?&?AGJM)E$$¿1E~DyIÃLÊ&L?)D6E$?A@E~DyIÃLz?&LFDG,D MAE®=C?)ÅE$E$FDM&LFAIJEÌ$LMA?<)@6=CBAIHÍ$EMAE)LGHFzD (#E~F&LFD º Ω ~D LUFzD?AF=C?)ÅE$@|DBa=C@6FAMAE
Rmº ÔZL 3=CFAÌ_DGH=CF
F MA &FAGHEÖMAE Ω M&LF)R
m @6E$<)@6$|E$FDEIÃL3=CFAÌ~D6GJ=CF?&?Ç-z?AE+I =F 6?A<)<>=C|EªL6|E +@|$KC?AIHGJÍ~@6E ºÔ¬E|Ì ÊA$ÐL)À Î <>=C?)@(Ì$Eª<A@6=BAIJÍ~ÐE+ ~Ì$@6G,Df6=C?AIJL93=C@6E+6?AG,ÅULFDE e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
(F(W n
i+1
)− F (W n
i )),
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
)),
L ÅCE$ÌSn
i+ 1
2
= maxp=1,...,m
(max
(∣∣∣λni ,p
∣∣∣ ,∣∣∣λn
i+1,p
∣∣∣))
= [λp]p=1,...,m6=CFD+IHE$ÑÅULIHE$?A@|Ñ<A@|=C<A@6E~+MAEIJL´´L1D@6GHÌ$E6LÌ$=BAGJE~FAFAEMAE F ¿ λn
i ,p@|E$6<aE$Ì
DG,ÅCE$E$FD λni+1,p
ÓÌ$LIJÌ~?AIJ~E$RI _D LUD W ni3@6E~6<aE$Ì~D6GHÅE$E$FDE I ~D L1D W n
i+1vE~D αn
i+ 1
2
?AF<&L@6LÐÍ_D@|EM)EÌ~=CFD@VSIJE º ÖF 6?A<)<>=C|EVz? GJI¬E!Ç)GH|D6E?AF ~DLUDGJFDE~@6$MAGJLGJ@|E V (W n
i ,Wni+1
)DE~I¬X?)E
F(W n
i+1
)− F (W n
i ) = A(V(W n
i ,Wni+1
)) (W n
i+1 −W ni
).
ÔZLÓÐLUD6@6GJÌ~E+MAEÀf=XE Àf=XE+@) Zz?&LFAM E$IHIJEÖE!Ç)GH|D6EªÅ$@|G&EE_Ç)LÌ_DE$E$FDfÌ$E_D6D6EÌ~=CFAMAG,DGJ=F ºËÖLFAfÌ$EªÌ$LI ~D LU<>E+<A@|$MAGHÌ~D6E$?A@MAE_ÅGHE$FD e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
A(V(W n
i ,Wni+1
)) (W n
i+1 −W ni
).
Á =CÐE IHE |ÈXDÍ$E 0 º m aE$|DZÊÈX<aE$@6Ba=CIHGJz?AE¿$GJI E~FA6?)GHDZz?AEIÃLÐLUD@|GJÌ~E A (V (W ni ,W
ni+1
))E$DvMAGÃLUKC=CF&LIHGJ6LBAIJE º ÖF FA=D6E λ1 ≤ λ2 ≤ ... ≤ λm
IJE~VÅULIJE~?A@6<A@6=C<)@6E$L6|=Ì~GJ$E~vL?ÇÅCE~Ì~DE~?A@6 <A@6=<A@6E~ r1, ..., rm ¿CE_D =FÐFA=UDE B = r1, ..., rm IÃLÖBAL6EMAE~ ÅE$Ì~D6E$?A@|®<A@|=C<A@|E$MAE
Rm º =CG,D R = [r1, ..., rm] IÃL´LUD6@6GHÌ$EMAE-<&L|LKCE´MAE IÃLB&L6E´Ì LFA=FAGJz?AE M)E R
m IÃL B&L6E B 3=C@6$E<&L@IJE~ÅCE~Ì~D6E$?A@|<)@6=C<A@|E$M)E A
(V(W n
i ,Wni+1
)) ¿GHI E~FA6?AG,Dvz?AE IJL´L1D@6GHÌ$E A
(V(W n
i ,Wni+1
))= Ri+ 1
2
Λi+ 1
2
R−1i+ 1
2
¿ = Λi+ 1
2
E$D-IÃL ´LUD6@6GHÌ$E MAE$ ÅULIHE$?A@|<A@6=<A@6E~+E~D R−1
i+ 1
2
E~|DªIJL´´L1D@6GHÌ$EGJFÅCE~@6|EM)EIÃL´ÐLUD6@6GJÌ~EVMAEV<&L66LKCE º G =CF Ì$=CFA|GJMAÍ~@6EvIHEÌ$=C<a=C@|D6E$E$FDMA?´|ÈX|D6Í$E+IJ=XÌ LIHE$E$FD®E~FD@6EÑIJE$ ´LUGJIJIHE$ xi
E_D xi+1¿XI ~DL<>E<A@6~MAGJÌ_DE~?A@
MA? |ÈX|D6Í$E º ^ ®MA=CFAF)E e
R−1i+ 1
2
W ni+ 1
2
=1
2R−1
i+ 1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
Λi+ 1
2
R−1i+ 1
2
(W n
i+1 −W ni
).
ÖF<a=C6E V ni+ 1
2
= R−1i+ 1
2
W ni+ 1
2
, V ni = R−1
i+ 1
2
W niE_D V n
i+1 = R−1i+ 1
2
W ni+1
=CF L?)@L
V ni+ 1
2
=1
2
(V n
i+1 + V ni
)−
αni+ 1
2
2Sni+ 1
2
Λi+ 1
2
(V n
i+1 − V ni
).
>Q>
DDI D 1D
ÆFD6E$F&LUFzDÌ$=C<)DEÑMAE+I LF&LI,ÈX6EÑM)? |Ì ÊA$ÐL )À* Î M&LUFAIJEÑÌ LIHGJFA$LGJ@|EfÅE$Ì_D=C@|GJE$I¿)M&LUFAIJE+Ì$L= λp 6= 0 ∀p ∈ 1, . . . , m =CF<aE$?)Df~Ì$@6GH@6E+MAEªÐLFAGHÍ$@6EªLUF&LIJ=KC?AE e
Q(V(W n
i ,Wni+1
))= Ri+ 1
2
αni+ 1
2
R−1i+ 1
2
,
L ÅCE$Ì
αni+ 1
2
=
Sn
i+12
|λ1| 0 . . . 0
0Sn
i+12
|λ2| . . .
ºººººº
0
º º º0
0 . . . 0Sn
i+12
|λm|
.
ËÖLFAfÌ$EªÌ$LIJE+6Ì Ê)$ÐL>)À* Î MAE_ÅXGJE$FD
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2Sni+ 1
2
Q(V(W n
i ,Wni+1
)) (F(W n
i+1
)− F (W n
i )),
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
)).
ÆF|GJ<AIJG8hLFD=CFL?A@6L
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2sgn
(A(V(W n
i ,Wni+1
))) (W n
i+1 −W ni
),
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
)).
» GJFALIJE~ÐE~FzDIJE+6Ì Ê)$ÐL>)À* Î E~|Df?AFAE+ÅUL@|GÃLFDE+M)?|Ì ÊA$ÐL>À Î z?AG¬ ~Ì$@|GHD e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2
∣∣∣A(V(W n
i ,Wni+1
))−1∣∣∣(F (W n
i+1) − F (W ni )),
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
)),
L ÅCE$Ì∣∣∣A(V(W n
i ,Wni+1
))−1∣∣∣ = Ri+ 1
2
∣∣∣Λ−1i+ 1
2
∣∣∣ R−1i+ 1
2
E~D ∣∣∣Λ−1i+ 1
2
∣∣∣ = diag
(1
|λp|
).
D tXopi 1r³sZt %ËÖLFAÐIJEBA?)DMAEz?&LIJG8&E$@vÌ~E|Ì ÊA$ÐLpFA=?A´LUIJIJ=FAvIJEÌ$=Ð<&LU@6E$@L ÅCE~Ì MAE~Ð|Ì ÊA$ÐLÐMAEÅC=CIH?AE$AFAGJE Ì LUÌ$E$-<a=C?A@IÃL96GH?AIJLUDGH=CF FX?)Ð~@6GJz?AEMAE$´|ÈX|D6Í$E$ ÊÈX<>E~@6Ba=CIJGHz?AE$~¿ L ÅC=CGH@-IJE;|Ì ÊA$ÐL ?&?ÇÌ$L@LÌ_D~@6GJDGHX?)E # » »Á ÏÒªÔ Á RQ=)¿(Ä =C? RQ@0¿M)=CFDIJED?&?ÇFz?AÐ~@6GHX?)E ~Ì$@|GHDf6=C?)fIJL93=C@6E+ÐLUD@|GJÌ$GHE$IHIJEf6?AG,ÅLUFzD6E e
gV FFC(V,W ) =1
2(F (V ) + F (W )) − 1
2sgn(A(U)) (F (W ) − F (V ))
>
: D D
B&L||?A@IHE$Ì$=?A@6BaE$fÌ L@6LÌ~D6$@|GJ|D6GJz?AE~ Xk(t)z?AGZ6=CFDMAE$f6=IJ?)D6GJ=CFA(M)E$Ñ~z?&LUDGH=CFAMAG8B
3$@|E$FDGHE$IJIHE$(|?AGHÅULFDE~ edXk
dt= λk(V (Xk, t)).
Ô¬E|Ì ÊA$ÐL # » »Á L9?AFAE KC@LF)MAE@|E$|6E$VBAIÃLF)Ì$EL^ÅE$ÌIJE6Ì ÊA~´L MAE Àf=XEM&LFAIJEÌ$LMAE$|ÈX|D6Í$E$ÊÈX<aE$@6Ba=CIHGJz?AE$ÊA=C=CKCÍ~FAE$~¿ZMA=CFDVIHE?&?ÇRFz?AÐ~@6GHX?)EÐ<aE$?)D $Ì~@6GH@6EMAE´IJL´LUFAGJÍ~@6E+6?)GHÅULFDE e
gRoe(V,W ) =1
2(F (V ) + F (W )) − 1
2sgn
(ARoe
)(F (W ) − F (V ))
E$F Ì Ê&LFAKE LFDIÃL=mÈE$FAF)EMAEÀ=E+<AL@(?AFAEÐ=^ÈE$FAFAEL@|GHD6ÊAÐ_DGHX?)E U =1
2(V +W )
º
ÖF FA=D6E~KLIHE$E$FDIJEª6Ì Ê)$ÐL# » À=XE ¹ » ÏERQR¿>Ï Î ) 0¿hMA=FzDIJE?&?XÇ FX?)Ð~@6GJz?AEE$DMA=CFAF)<AL@
gV FRoe (V,W ) = F (WV FRoe (0;V,W )) ,= WV FRoe (0;V,W ) MA~6GHKCFAE+IÃL6=CIH?)DGH=CF MA?-<)@6=CBAIHÍ$EÑMAEÖÀfGHE$ÐLFAF´IHGJFA$L@6GH66?AG,ÅLUFzD e
∂W
∂t+ A(U)
∂W
∂x= 0,
W (x, 0) =
V, x < 0,
W, x > 0,
E$F 6?A<)<>=C6LFDÑz?AEªIJLÓ|=CIJ?DGJ=F-MA?<)@6=CBAIHÍ$EÖFA=CF IJGHFA LUGJ@6EE$DÌ~=CFAFz?AE"I GHFA|DLFD tn ºÔ¬E|Ì ÊA$ÐL´Lv_Dª$Ì~@6GHDf<a=C?A@(IHE$<A@6=BAIJÍ~ÐE~(ÊzÈX<aE$@|B>=CIHGJz?AE~fÌ~=CFA|E$@|ÅULUD6G3~¿&E~Df<&L@IÃLv|?AGHD6ELv~D<A@|=CIJ=CF)KC+L?Ç-<A@6=CB)IJÍ$E$(F)=CFÌ~=CFA|E$@|ÅULUD6G3 Ä(Ï Î C ºÔ¬E+6Ì ÊA~´L)À* Î @|E~D@|=C?)ÅE+IJE+6Ì Ê)$ÐL# » À=EÖ<a=C?A@®IJE$(ÈX|DÍ~ÐE~ÊzÈX<aE$@|B>=CIHGJz?AE~(FA=CFÊA=C=CKCÍ~FAE$E$F E! >E~DfI _D L<aE+<A@6~MAGJÌ_DE~?A@ $Ì$@|GHD
W ni+ 1
2
= W ni +
1
2
m∑
p=1
γp
(1 − sgn(λp)
)rp
= W ni +
∑
λp<0
γprp
= WV FRoe
(0;W n
i ,Wni+1
),
M&LFA¬IHEÌ L¬=+I ~D L1D¬Ð=^ÈE$F V (W ni ,W
ni+1
)=
1
2
(W n
i +W ni+1
), E~D γp = t lp·
(W n
i+1 −W ni
) ¿<a=C?A@(<AIH?AMAEªMA~DLGJIH®ÅC=GJ@(<&L@6LKC@6L<AÊAE 0 º . º > º,=C?)@?)FAE-IJ=CG MAE-Ì$=FA6E~@|ÅULUDGH=CF 6Ì$LIÃLGH@6EÐ6?A@Ó?AF ÐLGJIHIÃLKCEÓ@|$KC?AIHGJE~@$¿IJE-6Ì Ê)$ÐL À Î Ì$=CE IHE$6Ì ÊA~´LU<# » »Á E~D<# » À=XE |=CFD´D6@6Í$<A@|=Ì ÊAE~ º ÆFE! >E~D$¿ |G=CF Ì~=CFA6GHMAÍ$@|EI $z?&LUD6GJ=CFMAEªÌ~=CFA6E~@|ÅULUD6GJ=CF6Ì$LIÃLUGJ@6E e
∂v
∂t+∂f(v)
∂x= 0
> =
DDI D 1D
= v ∈ RE~D f ?AFAE3=CFAÌ~D6GJ=CF@6~KC?AIHGJÍ$@|E M)E R
M&LFAR¿1IJE~E!Ç<A@6E~66GH=CFAMAE~ ?&?XÇFz?A$@|GJz?AE$
6=CFDfMA=CF)FA$E~<&LU@ e
gSRNHS(v, w) =
f(v) 6G f ′(u) > 0
f(w) 6G f ′(u) < 0
f(u) |G f ′(u) = 0
gRoe(v, w) =
f(v) |G f ′(u) > 0
f(w) |G f ′(u) < 0
1
2(f(v) + f(w)) |G f ′(u) = 0
gV FFC(v, w) =
f(v) |G f ′(u) > 0
f(w) 6G f ′(u) < 0
12(f(v) + f(w)) |G f ′(u) = 0
=
f ′(u) =
f(w) − f(v)
w − v|G w 6= v
f ′(v) 6G w = v
E~D u =1
2(v + w).
D tXopi 1r³sZt AgSRNHS(v, w) = gV FRoe(v, w) M&LFAÑIJEVÌ$LÖ=C?I _D LUDÑÐ=^ÈE$F u MA?|Ì ÊA$ÐL )À* Î E~|D<A@6GH~KLI u º Ô¬E |Ì ÊA$ÐL)À* Î VL=CFD@| ?)FAE®@|=CBA?)D6E$6|EE~D?AFAE®KC@LF)MAE E Ì LÌ~GHD <>=?A@IÃL6GH?AIÃL1DGJ=F;Fz?AÐ~@6GHX?)EÐMAEÌ$E~@|DLGJFAª<A@|=CBAIHÍ$E$ªÊÈ<aE$@|B>=IJGJz?AE~Ê)=CÐ=KCÍ$FAE~ º¹ LGHªMAE´LUFAGJÍ~@6ELFALIJ=CK?AEL?Ç+6Ì ÊA~´LZMAE À=EU¿ # »y»Á E~D# » Àf=XE z?AG|E BAL6E~FzDE$|6E~FzD6GJE~IJIJE~ÐE~FD6?A@-I LU<A<A@6= ÇGJÐLUD6GJ=CF MA?<A@|=CBAIJÍ~ÐEMAEÀGJE~´LF)F IJ=XÌ LUI<&LU@ ?AF<A@|=CBAIJÍ~ÐEIJGHFA LGH@6EU¿ Ì~E6Ì ÊA~´L<>E~?)DVM)=CFAFAE~@MAE~V|=CIJ?)D6GJ=CF)+FA=CFp<)ÊzÈX|GJz?AE$VE~DÐE~FAE$@" I LU<A<&L@|GHDGH=CFMAEÌ ÊA=XÌ$|DLUDGH=CFAF&LGH@6E~$¿LIJ=C@|z?AE+IÃL6=CIH?)DGH=CF <AÊÈX6GHX?)Ez?AGaÅC~@6GAE+I GJFA~KLIJG,DM E$FD@|=C<AGHE+E$|D?)FAE=CFAMAE´MAE MA_DE$FD6E¿ Ì~E$IÃLFA=C?A=BAIJGHKCE< LGJ@|E´?AF)E Ì$=C@|@6E~Ì~DGH=CF E~FzD6@6=C<)GJz?AE-L?pFAGHÅE L? MAE~<>=GJFD(|=CFAGJz?AE~ º
11 2IA6? =d 7@?$24 CK4 7:92 D? ;DC
ËÖLFAªI _D L<aE<A@|$MAGHÌ~DE~?A@ªMA?96Ì ÊA~´L )À* Î >¿³=CF L-E$|L ÈCMAED@|=C?)ÅE$@ª?AFAEÓBa=CFAFAEÓL<<A@6= ÇGJÐLUD6GJ=CF MAE IJL 6=CIH?)DGH=CFRL<A<A@|=Ì ÊA~E-MA? <A@|=CBAIJÍ~ÐEMAE-ÀGJE~´LUFAFpIHGJFA$L@6GH6<&L@VI E!Ç7<A@6E~6|GJ=CF e
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2sgn
(A(V(W n
i ,Wni+1
))) (W n
i+1 −W ni
).
> P
: D D
ÖF<)@6=1|E_D6DE W ni+1 − W n
iM&LFAvIJL B&L|E M)E$ÅCE~Ì~D6E$?A@|Ð<A@|=C<A@|E$ B = r1, ..., rm MAE-IJL LÌ$=BAGJE~FAFAE A Ì$LIJÌ~?AIJ~EªLvI ~D L1DÐ=^ÈCE~F³¿&=CFL?A@6L e
W ni+1 −W n
i =
m∑
p=1
γprp.
Ö@A(V(W n
i ,Wni+1
))= R sgn
(Λ(V(W n
i ,Wni+1
)))R−1,
L ÅCE$Ì sgn(Λ(V(W n
i ,Wni+1
))) E$|DfMA=FAFA$Eª<&LU@
sgn(Λ(V(W n
i ,Wni+1
)))=
λ1
|λ1|0 . . . 0
0 λ2
|λ2| . . .
ºººººº
0
º º º0
0 . . . 0 λm
|λm|
,
M&LFAIHE+Ì L= λk = 0 ¿A=CF<a=C6E sgn(λk) = 0 ¿AGJI³ E~FA6?)GHDfz?AE
sgn(A(V(W n
i ,Wni+1
))) (W n
i+1 −W ni
)= R sgn
(Λ(V(W n
i ,Wni+1
)))R−1
m∑
p=1
γprp
= R sgn(Λ(V(W n
i ,Wni+1
))) m∑
p=1
γpR−1rp.
Ö@ tlqrp = δpq=? lq E~|DIJE+ GHÍ$EÑÅCE~Ì~D6E$?A@Ì$=CIH=CFAFAE+MAEªIJL´LUD6@6GHÌ$E R−1 ¿&M =
sgn(A(V(W n
i ,Wni+1
))) (W n
i+1 −W ni
)=
m∑
p=1
γp sgn(λp)rp,
MAEª<AIJ?)f=FL e1
2
(W n
i+1 +W ni
)= W n
i +1
2
(W n
i+1 −W ni
)
= W ni +
1
2
m∑
p=1
γprp.
Ë L?)D@|Eª<&L@|D1
2
(W n
i+1 +W ni
)= W n
i+1 −1
2
(W n
i+1 −W ni
)
= W ni+1 −
1
2
m∑
p=1
γprp.
> @
DDI D 1D
Ë =
W ni+ 1
2
= W ni +
1
2
m∑
p=1
γp
(1 − sgn(λp)
)rp
= W ni+1 −
1
2
m∑
p=1
γp
(1 + sgn
(λp
))rp.
ÖF|?A<A<a=C6Ez? GHI³E_ÇGH|DE j ∈ 1, . . . , m DE~I¬z?AE
λjl < 0 < λjr
= λjlE$DÑIJL GJÍ~ÐEÅLUIJE$?)@(<A@6=C<)@6EMAEªIÃL6LÌ$=BAGJE~FAFAEªÌ LIHÌ$?AIH$EªLÓI ~DLUD W n
iE~D λjr
E$DÑIJL GHÍ$EÖÅLUIJE$?)@f<)@6=C<A@|EM)EIJLÖ6LÌ~=CBAGHE$FAFAEÌ LUIJÌ$?)IJ$ELÓI ~DLUD W n
i+1¿aMALFAfÌ$EVÌ$Lf GJFA|<AGJ@6LFD
MAE Î L@|D6E$F³¿ Î ÈXÐLF´E_DÔZL1Ç ÎÑÎ Ô4P =aE~D Î L@|D6E$FÐE~D Î ÈX´LF ÎÎ @ .>=F´@6E~Ð<)IÃLÌ~E λjl<&LU@
λjlE_D λjr
<AL@ λjrL ÅCE$Ì
λjl = λjl
λjr − λj
λjr − λjl
E~D
λjr = λjr
λj − λjl
λjr − λjl
.
ÆFA6?AG,DEª=CF I GJF1|E~Ì~D6EÖM&LFAfI E_Ç<A@|E$|6GJ=FMAE W ni+ 1
2
MAEªIÃL9L½~=CF6?AG,ÅLUFzD6E e
W ni+ 1
2
= W ni +
1
2
m∑
p=1,p6=j
γp
(1 − sgn(λp)
)rp +
1
2γj
(1 − sgn(λjl)
)rj
0 º =
E~D
W ni+ 1
2
= W ni+1 −
1
2
m∑
p=1,p6=j
γp
(1 + sgn(λp)
)rp −
1
2γj
(1 + sgn(λjr)
)rj.
0 º P
ÖF LM)MAGHD6GJ=CFAF)E 0 º = ®E_D 0 º P =F L?)@L
W ni+ 1
2
=1
2
(W n
i +W ni+1
)− 1
2
m∑
p=1,p6=j
γp sgn(λp)rp −1
4γj
(sgn(λjl) + sgn(λjr)
)rj.
=C?)LUIJIJ=FA®DE~|D6E$@fIJE+6Ì Ê)$ÐL>)À* Î >¿AD=?)DM LB>=@6M6?A@MAE~<)@6=CBAIHÍ$E$(ÊA=Ð=CKÍ$FAE~ º
> R
: D D
11 GK5@; 7) 7@5 4 ; EHG9? ;DCK5
65x)7Xoptt B i&q k -t i kÔ¬E$9$z?&LUD6GJ=CF);MAEC)LGHFzD (#E~F&LFD=CFA=XMAGJE$FA|GJ=CF)FAE$IHIJE$=M)$IJGHLFD I $Ì$=?AIJE~ÐE~FzD;MAEI E L?M&LUFA?AFÌ$LF&LI¬ ~Ì$@|GHÅCE~FD e
∂h
∂t+
∂
∂x(hu) = 0, x ∈ [−10, 10]
∂
∂t(hu) +
∂
∂x
(hu2 + g
h2
2
)= 0
= h E$DfIÃLvÊ&L?)D6E$?A@MAEªI E$L?³¿ u E$|DfIJLVÅG,DE~66EM $Ì~=C?AIJE~ÐE~FDMAEªI E L?E~D g = 9.81ms−2
I LÌ$Ì~$IH$@L1DGJ=F-MAEªIÃL<aE$LUFzD6E$?A@ ºÔZLÓÐLUD6@6GJÌ~E LÌ$=BAGJE~FAFAE+E$DÑM)=CFAFA~E<AL@
A(W ) =
0 1
c2 − u2 2u
= c =√ghº
Ô¬EM)E$?ÇÅULIHE$?A@|f<)@6=C<A@|E$fMAE A |=CFD u− c E~D u+ c ¿&IÃLvÐLUD@|GJÌ$E+M)E<AL66LKCEE$|DMA=CF)FA$E<&L@
R(W ) =
1 1
u− c u+ c
E~DfI GJFÅE$@6|EªMAEªIJE+ÐLUD@|GJÌ$EÑMAEª<&L66LKCEE$D e
R−1(W ) =1
2c
u+ c −1
−u+ c 1
.
ÔZLÐLUD@|GJÌ$E|GJKCFAEMA &FAGHEÑM&LF) IHE|Ì ÊA$ÐL )À* Î >¿XE$|D®E$FÐ<A@6GHFAÌ$GH<>EMA &FAGJEf<&L@ IJE~ _D LUD6Ð=^ÈE$FAMAEÓÀ=E Àf=XE+@ z?AG6=FzDMA=FAFA$E~<AL@ h =
hl + hr
2E_D u =
ul
√hl + ur
√hr√
hlhr
ºÁ E$<aE$FAMALFD´M&LUFAvIJE~DE$DÓFz?A$@6GHz?AE$v=CFL;|GJ<AIJE~ÐE~FzD<A@6GHIJE~v~D L1Dv=mÈE$FA h =hl + hr
2E_D u =
ul + ur
2¿d= hl
E~D ul6=FzDª@|E$|<>E~Ì~DG,ÅCE~ÐE~FzDªIJL´Ê&L?DE$?)@+E~D+IJLÐÅXGHDE~6|EvMAE
I E L?Ð6G x < 0 ¿ hrE~D ur
|=CFD@|E$6<aE$Ì_DG,ÅCE$E$FD®IÃLªÊ&L?)D6E$?A@ E~D IÃL+ÅG,DE~66EMAEÑI E$L?6G x > 0º
,=C?)@fÅULIHGJMAE~@(IJEª6Ì ÊA~´L>)À* Î E~DIÃLÓÐ=XMAG8&Ì LUD6GJ=CF´E$FD@|=C<AGJz?AEFA=C?A2LGH6=CFAMAE~fD6E$DFz?AÐ~@6GHX?)E$V|?A@VM)E$VÌ$LV=pGJIHªE_ÇGJDE$FDvM)E$V<a=CGHFzD6|=CFAGHX?)E$VE_DVF)=C?AVGHFD@6=XMA?AGH6=CF)IJLÌ$=C@|@6E~Ì~DGH=CFE$FD@|=C<AGJz?AE Ì$E FAGHÅE L? º ,=C?A@ Ì$E$IJL=F:<A@|E$FAM Ì$=CÐEÌ~=CFAMAG,DGH=CFGJFAG,DGÃLUIJEhl = 20m ¿ hr = 1m ¿ ul = 0 E_D ur = 0 t = 0.3s L ÅCE~Ì ?AFR´LGHIJIJLKCEMAE 100 <a=CGJFD6 &K?A@6E~ º @)¿ º R E~D®?AF´Ì$L D6E$D(L ÅCE~Ì%3=CFAMÐ6E$Ì &K?A@6E~ º ^)¿ º C
DDI D 1D
º ÆFA6?)GHDE®FA=C?ALIHIJ=CF)y@6E~<A@6E~FAMA@|E(X?)E$IJz?AE~E_ÇE$<AIJE~ DE~|D~<&L@$& º Ä ? ³LU@6M³¿& º ÏLUIJIJ=?AE~DE~D )º ¹ ºÎ $@LU@6M Ä(Ï Î RQ@XM&LFA ?AF6Ì ÊA~´Lf6GJ<AIHE<>=C?)@IHE$$z?&LUD6GJ=CFAMAE+)LGHFzD (#E~F&LFD<>=?A@ÅC=CGH@I E Ì LÌ~GHD6+MA?|Ì ÊA$ÐLÓ<>=C?)@(IÃL@6~6=CIH?)DGH=CFMAEªÌ$E~f<)@6=CBAIHÍ$E$ º #^'a#X7 X7f" !|@X7 \_# /1b, ´/1#X c*c*J%Öe GJIz LKCG,DyM ?AFv<)@6=CBAIHÍ$E M)EÀGJE$ÐLFMAE®Ì~=CFAMAG,DGJ=FGJFAG,DGJLIJE~ e
(h,Q)(x < 0, t = 0) = (1, 0), (h,Q)(x > 0, t = 0) = (0.3, 0)
FAEÓ! =CFAMAEªMAEªMA_DE~FzD6E|?ABA|=CFAGJz?AEE~D?)F;- Ì ÊA=XÌ|E3=C@6E$FD º #^'a#X7 X7V" _|@X7Ö\_#XE /1b,Ó\^7 ]ve Ô¬E~fÌ~=CFAMAG,DGH=CFAGHFAGHD6GÃLIHE$®|=CFDf´LGHFDE$FALFD e
(h,Q)(x < 0, t = 0) = (1, 0), (h,Q) (x > 0, t = 0) = (0, 0)
FAEÓ! =CFAMAEÑMAEªMA~D6E$FDEª6?)BA6=CF)GJz?AEªE~D?AF- Ì ÊA=XÌ+M LUÐ<AIHGHD6?AMAED6@6Í~2LUGJBAIHE$®|E3=C@6E$FD ºÖF Ì$=FA|DLUDEBAGJE~F z?AEVIHE$f@6~6?AI,D LUD6ÑFz?A$@6GHz?AE$f6?A@<AIJ?)6GJE~?A@6fÌ LUDE~|DÑ=CB)DE~Fz?AÑ<&LU@IHE6Ì ÊA~´L)À* Î 3=C?6# » À=E Ì$=CGHFAÌ$GHMAE$FDL ÅCE~ÌÌ~E$?Ç MA?|Ì ÊA$ÐLÐMAEÀ=E ºÖFRÌ~=CFA|DLUDEÐL?A|6Gz?AEÓIJEÓÌ LUIJÌ$?)I MAEIÃL LÌ~=CBAGJE~FAFAEE$F ?)DGHIJGHLFD+IJL=^ÈCE~FAFAEÐL@6G,DÊA DGHX?)ELU? IHGJE~?-MAEªIÃLvÐ=^ÈE$FAFAEªMAEÀ=E+FAEªÌ Ê&LUFAKCEª<&LIJE~(@6~6?AI,D LUD6fFz?A$@|GJz?AE~ º
: D D
−10 −8 −6 −4 −2 0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20sans correctionavec correction
A B ' ?A9 P BG /213 'I/213 5P1*1 > ( ) ' "
−10 −8 −6 −4 −2 0 2 4 6 8 100
2
4
6
8
10
12
14sans correctionavec correction
(6' "$"$ B G /2143 '0/213 U5 11 > ( ) ' "
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5sans correctionavec correction
K M ' P P G /213 'I/213 5P1*1 > ( ) ' "
−10 −8 −6 −4 −2 0 2 4 6 8 100
2
4
6
8
10
12sans correctionavec correction
8 <P (@' N P B G /213 '0/2143 5 11 > ( ) ' "
'
DDI D 1D
65x)7Xopt sx3tLÔ¬E$$z?&LUD6GJ=CF)-MAEIÃL M)ÈXF&LUÐGHX?)EMAE$-KL<>=C?)@-?AF ?&?AGJM)E<&L@ LUGHD ~Ì$@|GHÅCE~FDE~F:?)FAEMAGJE$F)6GJ=F-M E~6<&LUÌ$EV|=C?AIÃL93=@6E
∂W
∂t+∂F (W )
∂x= 0, x ∈ [0, 10],
W (x, 0) = W0(x).
,=C?)@23E~@6E$@(IHE+|ÈX|D6Í$E¿h=CFLm|=C?)D6E+I $z?&LUD6GJ=CFM ~DLUDfMAE$KL+<AL@ LG,D
p = (γ − 1)
[E − 1
2ρu2
].
Ô¬E$(ÅUL@6GJLBAIJE~(Ì$=CF)6E$@ÅULUDG,ÅCE$fE~DIJL 3=CFAÌ~D6GJ=CF-M)?6?&?XÇ <AÊÈ|GJz?AE6=CFDfMA=CF)FA$f<&L@ e
W =
ρ
ρu
E
, E~D F (W ) =
ρu
ρu2 + p
(E + p) u
= ρ @|E$<A@|$|E$FDEÖIJLMAE~FA6G,DMA? KLU¿ u E$|D®IÃLªÅG,DE~66EU¿ p E$|D®IÃL<A@6E~66GH=CF E_D E E$DI $F)E$@6KGJED=DLIJE º γ E$|D®IJEf@L<)<>=C@D®M)E$Ì Ê&LUIJE$?)@6®|<>~Ì$G8&z?AE$ º I>E~|D®6?A<A<a=C|ÖÌ$=CF)|D LUFzDE~D~KLI 1.4Ì$LM ?)FKL<AL@ LG,D(MAGJLUD=CGJz?AE ºÖF M)!&FAG,D$KLIHE$E$FD+I E~FzD6Ê&LIH<AGJEVD6=D LUIJE H <&L@ H =
1
ρ(E + p) E_DIÃL Ì$~IJ~@6GHD6vMA?96=CF
c =
√γp
ρ
ºÔZL LUÌ$=CBAGHE$FAF)EM)E F E$|DMA=CF)FA$Eª<&L@
∇F (W ) =
0 1 0
(γ − 1)H − u2 − c2 (3 − γ)u γ − 1
(γ − 2)uH − uc2 H − (γ − 1)u2 γu
.
ÔZLÓÐLUD6@6GJÌ~E ∇F (W ) E~|DMAGÃLUKC=CF&LIHGJ6LBAIJEU¿z6E$ÅULIJE~?A@6(<)@6=C<A@|E$ λj6=CFDfMA=CFAF)$E$f<&L@ e
λ1 = u− c, λ2 = u, λ3 = u+ c
E~Df6E~fÅE$Ì_DE$?)@6Ñ<)@6=C<A@|E$fL|6=XÌ$GJ~6=CFDfMA=CFAF)$f<&L@
R1(W ) =
1
u− c
H − uc
, R2(W ) =
1
u
12u2
E_D R3(W ) =
1
u+ c
H + uc
.
.
: D D
¾Ñ<A@|Í$+L Å=CGJ@E! >E$Ì~D6?AIJLÌ~=C@6@|E$Ì~D6GJ=CF E$FD@|=C<AGJz?AEGJFD@|=M)?AGHD6EM&LUFAÑIHE<&LU@LKC@6L<AÊAE0 º . º > _¿Ì$E|Ì ÊA$ÐL <>E~@6E~D-IJLp6GH?AIÃL1DGJ=F Fz?A$@6GHz?AE;MAEz?AE~IJz?AE$Ì$L-Ì$@|GHDGHz?AE$ E~F<A@6E~F&LFDt = 0.005 E_DIJLvÌ$=CFAM)GHDGH=CF-MAEª|DLBAGHIJGHD6Ñ~KLIJE 0.9 eÆF 3=CFAÌ_DGJ=F MAE~ÑÌ$=FAMAGHD6GJ=CF)GHFAGHD6GÃLIHE$IHE<A@|=CBAIHÍ$EM Æ?AIJE~@ÑLUMAÐE_DD6@6=CGHÌ ÊALÐ<)ÑÌ LU@LÌ D~@6GJDGHX?)E$$¿ IJE<A@6E~ÐGHE$@´E_D-IJE D@|=CGJ|GJÍ$E Ì Ê&L<A Ì LU@LÌ_D$@|GJDGJz?AE~ 6=FzD´Å@6LGJE$FD´FA=CFIJGHFA LGH@6E~ ∇λj ·Rj 6= 0 ¿A<a=C?A@ j = 1, 3 E_D<>E~?)ÅCE~FzD;~D6@6E+|=CGHD(M)E$=CFAMAE~MAEª@L@|!LÌ_DGJ=F³¿6=CG,DVM)E$V=FAMAE$MAEÌ ÊA=ÌU¿<AL@Ì~=CFD@|E´IJEÓMAE~?ÇGJÍ~ÐEÌ Ê&L<pÌ$L@LÌ_D~@6GJDGHX?)EÐE~|DIHGJFA$LGJ@|E ÐE~FDvMA$K$FA~@6 ∇λ2.R2 = 0 E_DÌ~=C@6@|E$6<a=CFAM ?)FAE MAGH6Ì$=FzD6GJFz?AG,D´M)E Ì$=CFDLÌ~Dvz?AG®6E<A@6=<&LKCEL ÅCE~ÌIJLVÅG,DE~66E u º=D6=CFAz?AE9<>=?A@Ì$E9<A@6=BAIJÍ~ÐE;IHE$@6~6?AI,D LUD6-Fz?AÐ~@6GHX?)E$ MA?6Ì ÊA~´L )À Î E~D IHE6Ì ÊA~´LÓMAE# » Àf=XE+6=CFDfIJE~;~ÐE~ º
Ci&uvk^tXu~k^n b,X7Ö\_#^'&7_ \^/1b& 5$#&7#" |% ~X]_ 0/1bÔ¬E+<A@6E~ÐGHE$@Ì Ê&LUÐ<Ì LU@LÌ_D$@|GJDGJz?AE+E~|D?AF)E=FAMAE+MAEª@LU@6!LUÌ~DGH=CF z?AG¬Ì~=CFDGJE~FD?AF<>=CGHFD6=CF)GJz?AE¿dÌ$E<A@|=CBAIHÍ$EVE$DÖD@6LGHD6 I LGHMAEMAEIÃLÐÌ$=C@|@6E$Ì_DGH=CF;E$FD6@6=C<AGHz?AEvGJFD@|=M)?AGHD6EM&LUFAIJE+<&LU@LKC@6L<AÊAE º . º?> º
~D L1DKCL?AÌ ÊAE x ≤ 5 ~DLUDMA@6=GHD x > 5ρl = 1 ρr = 0.01ul = 0 ur = 0pl = 105 pr = 103
º
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4Lan
i+ 1
2
∈]un
i ⊥uni+1, u
ni >un
i+1
[
VZ0& "4]T' .4: " A 52^0:Z3 "/3:52:S=/2K
f ′(an
i+ 1
2
) (un
i+1 − uni
)= f(un
i+1) − f(uni ).
J'/D 4Lun
i = uni+1
Pf ′(an
i+ 1
2
)= f ′ (un
i ):sn(xi, xi+1, u
ni , u
ni+1)
:3_')( `+8?5bq./ 0!352_3 "!:]g1')(*4\Y_4:0/\.:!0 "4:^
sn(xi, xi+1, uni , u
ni+1) =
1
∆xθ
∫ xi+1
xi
∫ tn+θ
tns(x, t, u)dxdt.
x "!&]& "!&9^'G(651.4>0>:13452c "!]''/ "Z "513_'/F "'!&. "LS!514RS!0 "&4T'/~3452]6&<=HJI 6D\;'NJ3 "'/!8./ 7'S.R-!&~Y "!'3 "M' A 51'/4/3i\ "F'/4/3 "F 52 @A :PS:~ "!T'/'/ $..i'/:~ !MYi0(*:3!3|Q !~4\9'/!:'G( "0&_0^ S9 @ :D0^:]3452)º
!#" $%!&'(*),+ -/.,),-01' )32546'7 4189(:425;<4>=?+@&258A4B.,)#-01' ),2*4641-C&
. ]9-!FRS! 0~'Z~3`'//3'//&4`T "51 @"A J'Z3452>6&<=HJIJ62V|4:R-!&9'/:Ug!& 3452e04::S4Px52/_'34:U.b!& `9U @ 1gc Q "^0B'/c3/57!&'N/ "O0: M' A 52:> " "51 @"A :P .5251U>'/ :39. " 0(4:qB0(*4:RS!/'/M3FQ !>'/: M' A 52:b`9":e#d "&0 0 S/S!³º &b'/c351/:1,+851'/P& "!F'/'/ 79"4:b:4:5b1!=!e`=3.. "P3!f& "!Z'/' "T 52')(*::!=0f:f3452!!c3 "M' A 51^C9_#d "0c0: "U./S!`9":f')(*::!T 5251/3JT'_4:5bB04:U4
&
!Z!& "M' A 51^ "51 @"A &D`9":D'=5 52:Z: "0./ /.N'/~º "!='/2 "e:0'/_3 "M' A 51_!9SK
∂u
∂t+∂u
∂x= −udz
dx
u(x, 0) = u0(x) =
ul
3x ≤ 20
ur
x > 20
z(x) =
03
x ≤ 20
13
x > 20.
E%)ºCK
v=3452_0T9" "'!516&<=HJI 67Y "!$:=& "M' A 51|(4:;0'N 52 A |3!9U2K
uni+ 1
2
= uni − 1
2
(un
i+1 + uni
)(zi+1 − zi)
un+1i = un
i − r(un
i+ 1
2
− uni− 1
2
)+ ∆tsn
i
4Lsn
i
:3c!&O3 `+&5bq./ 0O'/ 3. "51 @"A &0&F'G(4:.Qe "3::.:!0 "&4]
sni = − 1
∆x∆t
∫ tn+1
tn
∫ xi+1
2
xi− 1
2
u(x, t)dz
dxdxdt = − 1
8∆x
(un
i+1 + 2uni + un
i−1
)(zi+1 − zi−1)
:zi = z(xi).$4:/3 "7RS!2Q "!&f:b3 "M' A 52P'Nc "'!&. "'S/RS!14:UF!20/3 "U.S!4
0 : "U\|3.. "/=Uul
:~!4:.|S3524:0N3u∗ = ul exp(−1),
\|!& 8U.3
u∗:ur
0]9-.:3s = 1
ºYJ] 3.]RS!D3 "30ul = 6
:ur = 6 exp (−1)
P\.Z "'!&. "2f! 3:JV/ "/3º-~'/f:3|: "&3524=i'N^ "'!&. "1-!&524,?R-!&]0& "4:]'/_&452b6&<=HJIJ6 E 9" "| @ !3YºK!º :$`v'$4:5bT04:U4ZE%'4:5b0;h~:57j&0k;:vW lkRS!:k=m*hiW noPh| |W_n"pNKVZ0& "4]2K
un+1i = un
i − r(un
i − uni−1
)− r
2
(un
i + uni−1
)(zi − zi−1) .
. 9" 1RS!'3452 0&4:U.40hi5Dj0ke:1W_lk:RS!kee: "339L1'/c3.. " Y "!T:fT.:3DEd9 "/| @ !)º -"K_º$!.O " &&0
ul = 6:ur = 1
º| "!:O`bVc'G(651`:4L0! 4:5b :3 !&9"4: F'N : X:/0:a0O'/ "'!&. " S!5143/RS!`9"a'/ "'!&. " 'S.R-!&! 9"! 0L')(*4\2/U.:5140//3 Ed9" /2 @ !T)º*UK_ºi$& "!&'/' ">.:'/:_ "!3MQ:_0(3!3^0!a4:5b6&<=HJIJ6OY "!J'/73 "M' A 51cE))ºKZ\_Y "!_'G(4RS!q./ '/4`B.352 !:Y "!B "513B'G(651.40! 3452 6&<=HJIJ6 Y "!1'/: M' A 52:v "51 @"A &: "] "51 @"A g'G(/&3S t = 10s
ºC &'/| @ !3JE )ºoSK(-
%E * %% 2 *9**2 * *
"352RS!fRS!D'/B9-3D0&D S9 @ :>0!34526&<=HJIJ6eV '/15 51^Q "!&Z'` 52 @A \ " "51 @"A º&;\9^'ND9'!~0^'G(3!~Q "!&T'/ 3 "M' A 51 " & "52 @"A 1V7&#d43/!&1e:'/'20&! M' A 52B "51 @"A ºx~2R-!&i3 "!&9FRS!b'4:5b 6&<=HJIJ6 3F651`:aQ "!&F'/:F M' A 52:F " 52 @A :!&.UcR-!&OQ "!&'/:]& "M' A 51_ 52 @A :Pv @ 30U^'NF9-320&1: "U9": @ :20(* 03 1
2
Y "!^'/: M' A 52:T " "51 @"A & "5151 Q !T'/_ "51 @"A &º
"
0 5 10 15 20 25 30 35 403
3.5
4
4.5
5
5.5
6SRNHSanalytique
! #"%$&'(&)*,+.-0/21435 687+9'01:;3<+=:;333> (&)8'?"
0 5 10 15 20 25 30 35 402.5
3
3.5
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4.5
5
5.5
6decentreanalytique
@!AB C<?D& E"F$G<'(&)H,+I-J/2143<5 6K7<+L'J1:M3N"M+O:M3K33> K(G)8'P"
0 5 10 15 20 25 30 35 402
2.5
3
3.5
4
4.5
5
5.5
6Schema SRNHSSolution analytique
Q&# R<?DS.! N"&$'?(GK).S-J/21:+8'J1:M3T"M+VUW3K3 > (&)8'?"
−8 −7 −6 −5 −4 −3 −2 −1 0−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5 Non homogene homogene
y=0.5x
<YX Z<?D;"L\[ ]?M<P"\"?-P^<_]` bacedgfgah jiKM-lkm)h<(&"?-JK)$'(&)VS]'bkm) > ! n'
o X
%E * %% 2 *9**2 * *
! $%!&'(*),+ -4 &2 42*= '7 4189(:42 ;<4=:+ &258 4
. e: "0 A 3^'/_3 "M' A 52J!9S
∂u
∂t+
1
2
∂u2
∂x= −udz
dx
u(x, 0) = u0(x) =
ul
x ≤ 0
ur
3x > 0
z(x) =
zl
x ≤ 0
zr
3x > 0.
E%)º -K
! #"%$'&)(+*u(x, t) , u(x, t) *.-/$0(12*4365728/$9":57<;7-.-=*?>A@.B:C#(EDF":G/@H*JIK*.-/$
(12*L-=57DF(E$9":57M-/$N;#$9":57+;#"%@H*O #(QP@H5#R)DSG/TU* ,JVWSX I'-/" ∂u(x, t)∂t
= 0 W68 "
u(x, t)!&>#d ":/ "a:kB4 @ !'/ A Pr\ u(x, t) :3]!>3 "'/!&/ "OV/ "&/3D0!
M' A 521E%ZY -K\P' "~ "1
2
∂u2
∂x= −udz
dx⇐⇒ ∂
∂x(u(x, t) + z(x)) = 0 ⇐⇒ u(x, t) + z(x) = c
4LcVZ!&f: "VU.[Y
S\] #"%$U&)(_^ (E`-=8.a+B/Tb;c@H*.-JP+*.8/$J*UPZ;#@%3d;#"%$e*/TU*/+$fDS*g8?;7-M-/$:;#$":57+;#"%@H*P+57(1@4DS*hP@H5#R)DSG/TU* ,JVWSX Ii-/"
uni + zi = c ∀ (i, n) ∈ Z × N
*/$zi = z(xi).
E%ZY "KjOkml_n6 % 9oprq57(-sDt;`8.57 #"%$9":57u E*gDt;< #"S-=8/@HB/$9"S-);#$":57u #(`$e*/@vTU*w-=57(1@H8.*K x;#Z-D?^yB/$N;/P+*i8.57@/@H*.8/$J*/(1@O E*0D?^ (E2*O E*.-i E*/([z3d;|[57Z-~-/(E"%7;#+$e*.-O
i) sni = − 1
4∆x
(un
i− 1
2
+ uni+ 1
2
)(zi+1 − zi−1)
ii) sni = − 1
8∆x
(un
i−1 + 2uni + un
i+1
)(zi+1 − zi−1)
57un
i+ 1
2
*.-/$D?^yB/$:;#$ EB+"_D?^yB/$N;/P+*hP@HB? #":8/$J*/(1@HDS*L-=8.a+B/Tb;Qqq@H*.-JP+*.8/$e*PZ;#@3d;#"%$J*/TU*/+$DS*f8?;7-~-/$:;#$":57+;#"%@H* WjOkEp =^P'_4:5bB0_9" "'!51TT(43=0^'N752 A 3_!9S
uni+ 1
2
=1
2
(un
i+1 + uni
)− 1
2sgn
(f ′(an
i+ 1
2
)) (un
i+1 − uni
)
+ ∆x
2
˛
˛
˛
˛
f ′
„
an
i+12
«˛
˛
˛
˛
sn(xi, xi+1, u
ni , u
ni+1
)
un+1i = un
i − r(f(un
i+ 1
2
)− f
(un
i− 1
2
)))
+ ∆tsni
E%ZY*oUK
U
`9"
sn(xi, xi+1, u
ni , u
ni+1
)= − 1
∆xθ
∫ xi+1
xi
∫ tn+θ
tnu(x, t)
dz
dx(x)
= − 1
∆x
(un
i+1 + uni
)
2(zi+1 − zi) ,
4L'/_5 A .3 θ VJ04\v0'/_351/T_\Dmh~:(-qpY. f ′(an
i+ 1
2
)=
1
2
(un
i + uni+1
),:
f(un
i+ 1
2
)− f
(un
i− 1
2
)=
1
2
(un
i− 1
2
+ uni+ 1
2
)(un
i+ 1
2
− uni− 1
2
).
$e351'NU`P e!.
uni+ 1
2
=1
2
(un
i+1 + uni
)− 1
2sgn
(f ′(an
i+ 1
2
)) ((un
i+1 + zi+1) − (uni + zi)
)
un+1i = un
i − r
2
(un
i+ 1
2
+ uni− 1
2
)(un
i+ 1
2
− uni− 1
2
)+ ∆tsn
i .
$e!&/'/.S|'N7:'N/ "E%ZY*UK\P')(*4\Y_340:.:!0:9-U
uni+ 1
2
=1
2
(un
i+1 + uni
)
= uni+1 +
1
2(zi+1 − zi) .
u(x, t)37! '/!&/ " 3q./ 1~:3B3!'51U>
un+1i = un
i
P'i:(*:!BRS!'G(4:.Q_: "3\.!&Z0!&452b6&<=HJIJ6 E%ZYoSK~VZ4:RS!9'U.^g
r
2
(un
i+ 1
2
+ uni− 1
2
)(un
i+ 1
2
− uni− 1
2
)− ∆tsn
i = 0
:4RS!9!&=g
sni =
1
2∆x
(un
i+ 1
2
+ uni− 1
2
)(un
i+ 1
2
− uni− 1
2
)
= − 1
4∆x
(un
i+ 1
2
+ uni− 1
2
)(zi+1 − zi−1) ,
0( 4Le'/7:52 A 3_0/34\../ "0 sni
Y='/'!i0^ "
uni+ 1
2
=1
2
(un
i+1 + uni
) P'/ "3
uni+ 1
2
+ uni− 1
2
=1
2
(un
i+1 + 2uni + un
i−1
),
/':(*:!&='N>0:!8+8 A 51_034:/.q./ 0!352^ !:]0'G(4:.Q_: "3\.!&)Y
"
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! .,),8 '(:),+ ->&;<025),%!&4
HZ "!&$'/' ".:'/:: "!MY0(*:3!3$Q "!&'/i4:5b^68<ZHZI 6f\$:'/!-0!Z?34520|W8?v040!&$W_lk:RS!k]mhiW_no&P"h| |W n"pY "!'/J57!'/. "fS!5143/RS!T0: M' A 52:T " "51 @"A &)Y~f3452B(430^'/B52 A 3 !9U.!K
un+1i = un
i − r(g(un
i , uni+1
)− g
(un
i−1, uni
))− ∆tsn
i ,
`9"g(un
i , uni+1) =
1
2
(1
2
((un
i+1)2 + (un
i )2)−∣∣∣f ′(an
i+ 1
2
)∣∣∣(un
i+1 − uni
))
:
sni =
1
4∆x
([1 + sgn
(f ′(an
i− 1
2
))] (un
i + uni−1
)(zi − zi−1)
)
+1
4∆x
([1 − sgn
(f ′(an
i+ 1
2
))] (
uni+1 + un
i
)(zi+1 − zi)
).
. e30 5251_ "&0/ "N'
u0(x) =
33
x ≤ 0
33
x > 0
:=!c#d "0c0 S/S!
z(x) =
03
x ≤ 0
13
x > 0. n2 n) Hnd = 9 kYm*vXW_npx "!&7'/F& "M' A 5120&ch @ :P " &&0 !!'./ " 0(*52&'/!0B. A >Y:..2R-!&F& " @ b`9F'/c9-3 u P &U> "51511: "0. "]/.N'/J'/f "&0/ "!9U.dY
u0(x) =
1 − z(x) + ε
0.1 < x < 0.2
1 − z(x)/& "
,
z(x) =
0.25(cos(π(x− 0.5)/0.1 + 1) |x− 0.5| ≤ 0.1
03/ "
,`9"e "ε = 0.001
P$3 "ε = 0.2
Y;J110:!8+ `B'/c3452 6&<=HJIJ6 \b'Z?4:5bL0&Fh~:57!0&kF\DW_lkRS!k SD0 "&4F'/:D5 51]43!'..fS!524:R-!&cE 9" " @ !3 vZY 2\ vZY*UK=\_'NB98.:>0&7 "U9 @ :2:3_0D'G( "03D0 0.78
RS!VZ'/>5 52_RS!^:'/'_ "M&.:S!f0'/_ "51 @"A &#Y
('
−10 −8 −6 −4 −2 0 2 4 6 8 101.8
2
2.2
2.4
2.6
2.8
3
VazquezSRNHSAnalytique
Q R<?DS.! N"&$'?(GK)
u-J/21:+V'J1:h"]+\:;33 > K(G)8'?"
−8 −7 −6 −5 −4 −3 −2 −1−6
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5SRNHSVazquez
= o <?M"Z\[ ]??]<P"S"?-P^<_M` aced f4a]'O "-P^_]` 8<N-0/\1:+ 'J1:5 3<+ > ])8'?1 35%W
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Q R<?DS.! N"&$'?(GK)
u]' km) h'0143<58"]+V-0/\1:+
:;33 > K(G)8'?"
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
h# RK<S<.! N"&$'(&)
u-0/\1:+V'01435"M+\:M3K3 > (&)8'?"
o
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:<; : t H D swfs B\zs @^z\uED D|tswfs Fw wIGCHsJD
. e: "0 A 3^'/_3-V A 51]S-Y3MQ "'/RS!_ e'/4/3J " "51 @"A >C !9UK
∂W
∂t+∂F (W )
∂x= Q(x,W ), (x, t) ∈ D × R+, D ⊂ R
W (x, 0) = W0(x), x ∈ D
E%ZY K
4LW : D × R+ → Ω
V7!9\.!&>gm
052: "]0bRS!S4D: "3394:P$'/:9/M'/:~0(4:.`P- 5251J'/D5bP&'/fRS!U..4 0& 52 !&9":52:S|:T')(*4: @ /Z0|'J0!3 "M' A 52Z0_'Nf524:`RS!J0:|!0T\T'N]!&!T\T'NDRS!S4 0 51 "!&951U0'/$0!f& "M' A 51$0i6&SV?XW;:U`P Ω 4:Sx!] "!&93xMY "4;0
RmY|#d "\. "
F:3>0&4:F0
Ω0
Rm0&.1#d "\./ !8+RS!F'G( " 3!Y "F3k4 @ !' A 32\
Q(x,W )'/ 352_3 "!:#Y
x "!&=:^`='/ 3452F68<ZHZI (43=0^'N752 A 3_!9S!K
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
(F(W n
i+1
)− F (W n
i ))
+αn
i+ 1
2
2Sni+ 1
2
∆xQni+ 1
2
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
))+ ∆tQn
i , E%ZY "K 4LSn
i+ 1
2
= maxp=1,...,m
(|λn
i p|, |λni+1p
|) 1L
[λp]p=1,...,m
"U'9'/!& "&0~'/Z52.3/ M/:_0
FPλn
i p
:λn
i+1p
3 "U=`'!'4T33Q::.9":52:SJg>'G(4:.W n
i
:W n
i+1
P\αn
i+ 1
2
!15 A .3T0 "U. "'/P Qni+ 1
2
:3'G(*3 `+&5bq./ 70'N_V.T "1 "51 @"A &0')(*4\Y_4:0/\.:!0'/ 9 "'/!&52_0^: "U.'/
[tn, tn + θ[×[xi, xi+1[m*hi(-pY
. 3!Y "1RS!(*'\+83.1!4:.^/U.:5140//3 V (W ni ,W
ni+1
) Pr';RS!B'N52.3/D0<= - 9"4:^'N>:'N/ "
F(W n
i+1
)− F (W n
i ) = A(V(W n
i ,Wni+1
)) (W n
i+1 −W ni
).
$(*4\Y^4:0/\.!&=0&:9-/:SZ' "
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
A(V(W n
i ,Wni+1
)) (W n
i+1 −W ni
)+
αni+ 1
2
2Sni+ 1
2
∆xQni+ 1
2
.
~ "5152_ "B!&Q "34^R-!&f'_3-3 A 51FE%ZY K~3=U-Q:MY "'/RS!P'/ "3T "cQ:!&=43/3A(V(W n
i ,Wni+1
)) 0^'N>52 A J!9SA(V(W n
i ,Wni+1
))= Ri+ 1
2
Λi+ 1
2
R−1i+ 1
2 4LΛi+ 1
2
P Ri+ 1
2
\ R−1i+ 1
2
"U33Q::.9":52:S'NZ52/:;09'!3x& "3:P'/Z52.3/0 . @ :~'N]5b=0J. @ /U93_`'!'4ZgD'G(4:. V (W n
i ,Wni+1
) P&0( 4LF!&4:.Q^340:!RS!v(4:IK
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)−
αni+ 1
2
2Sni+ 1
2
Ri+ 1
2
Λi+ 1
2
R−1i+ 1
2
(W n
i+1 −W ni
)+
αni+ 1
2
2Sni+ 1
2
∆xQni+ 1
2
"n
E*9 #%% %
⇐⇒ R−1i+ 1
2
W ni+ 1
2
=1
2R−1
i+ 1
2
(W n
i+1 +W ni
)−αn
i+ 1
2
2Sni+ 1
2
Λi+ 1
2
R−1i+ 1
2
(W n
i+1 −W ni
)+αn
i+ 1
2
2Sni+ 1
2
∆xR−1i+ 1
2
Qni+ 1
2
.
E%ZY 'K ]'/B^02V-3 A 52:D 52 @A :] c! "!89"^')(*,+&&3 "0!5 A .3B0 "U"'
αni+ 1
2
= Sni+ 1
2
|Λi+ 1
2
|−1 Y. e351'N:_0fE%ZY 'K\P')(*4\Y_4:0/\.:!0:9-/:U
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2Ri+ 1
2
|Λi+ 1
2
|−1Λi+ 1
2
R−1i+ 1
2
(W n
i+1 −W ni
)
+1
2∆xRi+ 1
2
|Λi+ 1
2
|−1R−1i+ 1
2
Qni+ 1
2
,
"5151sgn
(A(V(W n
i ,Wni+1
)))= Ri+ 1
2
|Λi+ 1
2
|−1Λi+ 1
2
R−1i+ 1
2:|A(V(W n
i ,Wni+1
))|−1 = Ri+ 1
2
|Λi+ 1
2
|−1R−1i+ 1
2
,
'/ T'/_345226&<ZHZI 6Q !T'/_3 "M' A 511E)ZY "K~(4:
W ni+ 1
2
=1
2
(W n
i+1 +W ni
)− 1
2sgn
(A(V(W n
i ,Wni+1
))) (W n
i+1 −W ni
)
+12|A(V(W n
i ,Wni+1
))|−1∆xQn
i+ 1
2
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
))+ ∆tQn
i .
E%ZY "K
k ='N7&./RS!_ "Q:!&=:03]')(*4\51 C":m*^|n"p
V(W n
i ,Wni+1
)=
1
2
(W n
i+1 +W ni
).
#" .,),8 '(:),+ - & =:80;<' +@&2 .,4 25+ .;<4 4!' )#-@( 41-' - ( '7 4189(:+ + 25' ),4>)32*2*0p &.,)254
HZ "!&7''/ "_!8./'/3_'/1&452L6&<=HJIJ6 Q !f'/3/57!&'N/ "OS!5143/RS!10!3 "M' A 510J6&/UV?XW;UC9Z!1#d 0234 @ !'/:04Y&0U~0&Z'/J9NM'/T3./'/ x :0 "&4z(x)
Ph(x, t)
3343UU~'N !8.!&0&Z')(*!B:ζ(x, t) = h(x, t)+z(x)
'/^3!3#%'/MP
u(x, t):3x'NT98.:0'G(`!]:
Q(x, t) = h(x, t)u(x, t)V'0&4M`PC'G(4RS!q./ ]0&
6&SV?XW;:UiC9. "Y @ T/34 @ !'/ A ~(430&:Z "!;'/_#d 51T!9SK
∂h
∂t(x, t) +
∂(hu)
∂x(x, t) = 0
∂(hu)
∂t(x, t) +
∂
∂x(hu2 +
1
2gh2)(x, t) = −gh(x, t)∂z
∂x(x).
E%ZY n"K
'
%E * %% 2 *9**2 * *
h(x,t)
z(x)
u(x,t)
fond de la rivière
Profil du lit de la rivière
68 "W (x, t) =
(h(x, t)
hu(x, t)
) !&F '/!&/ "3.. "B0aE%ZY*nUK^(*:3V?gq?X0/3W
V3kT34 @ !' A i943U ∂W
∂t(x, t) = 0
:u(x, t) = 0
P0T'/J0!-+& A 51|4RS!q./ 0!3-3 A 51FE%ZY n"K~3=4:R-!&9'/:U.]g1
2
∂(gh2)
∂x(x, t) = −gh∂z
∂x(x) ⇐⇒ h(x, t) + z(x) = c.
Z b #"%$&)(+*0DS*~-=8.a+B/Tb;Q#B/@/" *LDt; C− P@H5.P@v":B/$eBL-/"DS*L-=8.a+B/Tb;U;/PxPDF"9&)(+BOU(E2*L-=57DF(1$9":57 -/$:;#$":57+;#"%@H*4@H*/$9@.57(E#*hn
i + zi = c,*/$
uni = 0 ∀(i, n) ∈ Z × N,
jOkml_n6 % 9oprq57(-sDt;`8.57 #"%$9":57u E*gDt;< #"S-=8/@HB/$9"S-);#$":57u #(`$e*/@vTU*w-=57(1@H8.*K x;#Z-D?^yB/$N;/P+*i8.57@/@H*.8/$J*/(1@~-=57(x-0D?^ (12*i E*.-0 E*/(#z3657@vTU*.--/(E"%7;#+$J*.-ii) sn
i = − g
4∆x
(hn
i+ 1
2
+ hni− 1
2
)(zi+1 − zi−1)
ii) sni = − g
8∆x
(hn
i+1 + 2hni + hn
i−1
)(zi+1 − zi−1)
DS*L-=8.a+B/Tb;Qqq#B/@v" *4Dt; C P@H5.P@/":B/$JB WjOkEp68 "
W (x, t)!&=3 "'/!8./ >3q./ #YJ "&
u(x, t) = 0P0;;'/|V8V A 52DE)ZY*nUKVZ4:RS!9'UZg
∂F
∂x(W (x, t)) = Q (x, t) ,
`9"W (x, t) =
(h(x, t)
0
) PF (W (x, t)) =
(0
12gh2(x, t)
) : Q (x, t) =
(0
−gh(x, t) ∂z∂x
(x)
).
vT9'/:!3 ": "U=0 "&4=λ1 = −c \
λ2 = c
'8
E*9 #%% %
:='N752/:_0^3. @ R :='N752/: 0^3. @ ]U9"3 R−1 3 "U=0 "4:=
R =
(1 1
−c c
) \ R−1 =1
2c
(c −1
c 1
),
`9"c2 = g
hni + hn
i+1
2
Yx "!&=:^`='G(4:.QJ4:0/\.!&=:(*4:3
W ni+ 1
2
=
hni + hn
i+1
2
− c2
(hn
i − hni+1
)−(
zi+1−zi
2
)
:='G(4:.Y_ "3\.:![hn+1
i
qn+1i
]=
[hn
i
qni
]− rg
2
0
(hni+ 1
2
)2 − (hni− 1
2
)2
+ ∆tQn
i
E)1YC"K`9"
Qni =
[0
sni
].
~ "5152'/| "'!&. " :33.. "/P'/ "3W n+1
i = W ni
\hn
i+1−hni−1 = − (zi+1 − zi−1)
P/':(*:!&=RS!f'/70!-+& A 51^ "51Q .U._0&bE%ZYCUK~V=4RS!9'U.]g
sni =
g
2∆x
(hn
i+ 1
2
+ hni− 1
2
)(hn
i+ 1
2
− hni− 1
2
)
=g
4∆x
(hn
i+ 1
2
+ hni− 1
2
) (hn
i+1 − hni−1
)
= − g
4∆x
(hn
i+ 1
2
+ hni− 1
2
)(zi+1 − zi−1)
0( 4Le'/7:52 A 3_0/34\../ "Y.
hni+ 1
2
=1
2
(hn
i + hni+1
) P/'r(3!=RS!1
2
(hn
i+ 1
2
+ hni− 1
2
)=
1
4
(hn
i+1 + 2hni + hn
i−1
),
0( 4Le'/70!-+& A 51^0/34\../ "Y
+@; ' 2*' )#=?+ -]& =:8 0;<' 9' 7 418 ' &(?2541= =?80;<'@=
$Z 0RS!'/>'4:5b 6&<=HJIJ6 '|VbS43Ub0&: "51:>.O#d "5D!'N/ "`9"f!e!&.3^4:5bBQ:3#d "35bS0&4:9":'/ "&Q4]Q !'N757!'/. "FS!514RS!^0= q?M' A 52:i& "b 52 @A :)Y-$b\+8' ".U'G(651`:4=0&!b&452D04:U.34 51 "U0Z-Y
'4-
%E * %% 2 *9**2 * *
<= -PW8?v:P_YtYYPvY "!^'N57!'/. "LS!514RS!B0] M' A 52:]U-YMY "'/RS!^ q?52 @"A :FEd M' A 52:]0&D514R-!&]0]!/0&K^h~:57j0&kb\fW lkRS!:kLmh| |W_nPhiW_noPUWZ|n"npY "U~& "Y "4=!Z0/334:../ "B04::S4:_0Z'/]3/ "2 "51 @"A &=\ "UZ/3 "M&S!!&e34521YV#d "52UY "!T^-Q^0]3-V A 51#YQ~^&4521:(*4:!^:!'/^4\Y^0^'N7#% K
W n+1i = W n
i − ∆t
∆x
(FRoe
i+ 1
2
− FRoei− 1
2
)+ ∆tQn
i
4LFRoe
i+ 1
2
V'N #d "\./ >0!1!8+BS!514RS!=g^')(*3.Utn!B9"`!20')(*SV#%
i+1
20 "&4]FRoe
i+ 1
2
=1
2
(F n
i+1 + F ni − 1
2
∣∣∣Ai+ 1
2
∣∣∣(W n
i+1 −W ni
))
4L Ai+ 1
2
= Ri+ 1
2
Λi+ 1
2
R−1i+ 1
2
V|'/f52.3/0J<= -J\ ∣∣∣Ai+ 1
2
∣∣∣ = Ri+ 1
2
|Λi+ 1
2
|R−1i+ 1
2
PΛi+ 1
2'N52.3/D0_9'/:!^3 "3P Ri+ 1
2
'/5b70B. @ B: R−1i+ 1
2
'G(/U9":B0B'/5bq.=0J @ P8RS!r04:Q:00Z0!8+b4:..9" " W n
i
:W n
i+1
Y8vZ..51Ui0!.:51]3 "!3^ Q "34^(43=0^'ND#% "c!&9U.!K
Qni =
1
2
(QL
i− 1
2
+ QRi+ 1
2
),
`9"QL
i− 1
2
=[I + sgn(Ai− 1
2
)]· Qn
i− 1
2:QR
i+ 1
2
=[I − sgn(Ai+ 1
2
)]· Qn
i+ 1
2
4LIV_'N15bq.^0(/0:U.4D\ Qn
i+ 1
2
= Q(xi+ 1
2
,1
2
(W n
i +W ni+1
)) &43U.>!&3 `+&5bq./ F0!c.352^3 "!:]gB')(*U.V#%
i + 12
Y$3. @ `U=\]4:3.!&P e!.
W n+1i = W n
i −∆t
∆x
(ΦV az(W n
i ,Wni+1,Qn
i+ 1
2
) − ΦV az(W ni−1,W
ni ,Qn
i− 1
2
))+
∆t
2
(Qn
i+ 1
2
+ Qni− 1
2
)
`9"
ΦV az(W ni ,W
ni+1,Qn
i− 1
2
) =1
2
(F n
i+1 + F ni −
∣∣∣Ai+ 1
2
∣∣∣(W n
i+1 −W ni
)− ∆x sgn(Ai+ 1
2
)Qni+ 1
2
)
v]345226&<ZHZI 6(4:=3 "!'N7#d "352 3!9U
W n+1i = W n
i −∆t
∆x
(ΦSRNHS
(W n
i ,Wni+1,Qn
i+ 1
2
)− ΦSRNHS
(W n
i−1,Wni ,Qn
i− 1
2
))+∆t˜Qn
i
`9"
ΦSRNHS(W n
i ,Wni+1,Qn
i+ 1
2
)= F
(W n
i+ 1
2
)
'
E*9 #%% %
`9"
W ni+ 1
2
=1
2
(W n
i +W ni+1
)− 1
2sgn
(An
i+ 1
2
) (W n
i+1 −W ni
)− 1
2|Ai+ 1
2
|−1∆xQni+ 1
2:˜Qn
i =1
∆x∆t
∫ xi+1
2
xi− 1
2
∫ tn+1
tnQ(x, t)dxdt
VZ!&D `+8/52/ "F0&]'/7V./_& "e& "52 @"A _0'G(4:.Q_: "3\.!&)Y 1'NL34:.!3c0! 3452 68<ZHZI 6 :1'/34520&W lkRS!:ke\bhi5Dj0kP; ":5b3RS!bRS!(/'$ e!&b0[r4:&b0D'/52 A 310&2.^'/B.:5123 "!3b\7'/#d "\./ 0&!!-+cS!524:R-!&P'/:T0!-+4:5bB^ "U=0 "&]4RS!9'U.:)Y
!(:41-=:),+ -& =?80;<' &- + 2254B01.,47 0
'L`0(*!& V-3 A 52OU-YMY "'/RS!L'/4/3 "51 @"A Pi'/e4:5b 6&<=HJIJ6 3.5 A &B!a4:5b'/3/RS!>04::S4251 "U`Pr::$ "&517M/:aR-!&1:>4:5bcV0( "03a! .:52c\c 3[Y=x "!0&c`c 52 @A :.R-!&34:UU0bQ "U.1 "RS!2L3452e "U9": @ 293F'N "'!&. " :U. /RS!Pi0 /'=V4:3.O0(/U.3 80&!/3! "3\. " U "RS!OQ !b'/'/B "M&' A 51#Y B'b`>0'NL52&F/'i\+8/V.0&>'N3 "'/!&/ "U83/RS!c!0&/: "U.-!&.4b0 "U.:D3q./ >RS!(/'; "U9-/:U70 @ 30]UfR-!&b0 S/S!.4P!&.351U0'(VZ/0RS!4_0:]0_9" "!&'/ "T51'/:|\.]0 S/S!.4 =!& "0e004:U.P~52/>'| a! & "M' A 51cR-!&=Y33.e!1'Na '/!&/ " S!514RS!! 9"`! 0L'N0/3 "U/S!40! #d 0 Ed9 "/1 @ !3 TZY -o : TZY -UKvY'J4:4f::./]0_#%_'G(\+-3/ "0!4:5b26&<=HJI 6cgB'G( "03_0!-+fQ "!&043^\.>0& 51!'.4[Y &_ .37`P' '/51.:!3 "U^/U.3 80!&.J!O9"`!O0'G(4:.Q>&40&/:!^Y "! '_9/M'J&S-3/RS!FE% "a .
V'/79:!]0:^93NM&'/
U-/RS!:J0^: "52Y "U. t (v1, . . . , vm)K|0^'ND#%: "e3!9U K
(vp)+i = (vp)
ni +
1
2φ(θp)
((vp)
ni+1 − (vp)
ni
)
:(vp)
−i = (vp)
ni − 1
2φ(θp)
((vp)
ni+1 − (vp)
ni
)`9"
θp =(vp)
ni − (vp)
ni−1
(vp)ni+1 − (vp)n
i
(vp)
ni+1 6= (vp)
ni
0'_` 4Lλn
i+ 1
2
> 0\
φ(θp) =
0
θp ≤ 0
θp
30 ≤ θp ≤ 1
1
θp ≥ 1.
'o
%E * %% 2 *9**2 * *
$!. " 4:37'D93NM&'/f "&V9.9"B: #d ":/ "0793NM'fU83/RS!:1\0:f'G(4:Q_340:!0\98UCK
W ni+ 1
2
=1
2
(W−
i+1 +W+i
)− 1
2sgn
(Ai+ 1
2
) (W−
i+1 −W+i
)
+∆x
2
∣∣∣Ai+ 1
2
∣∣∣−1
Qni+ 1
2
(xi, xi+1,W−i+1,W
+i )
4L Ai+ 1
2
3='/J3 - M/:_`'/!&'/4^>')(*4\0]< 8_:#d :/ "c0V −
i+1
:V +
i
Y
01=:&. (*'(:= >&;<025),%!&41=
jOkg 9nxp=Ml k 0 2 f2`:Px ".:3.2'NcQ:3#d "35bB0! 3452O6&<=HJIJ63!D! 4: "!'/:52:U7gc9?/. "a.0#YJ(V7!.:3f "Y "342DvXW;RS!m*vXW n"pGP:!.B.:3.42f68:,+ : !+emW_6&(-Cp`9":D'f3452b;H. / ;H . :9"4:UJ'/b?X 4:.4]\+&\.[Yr$:[Q:](V]!a "M&' A 517 V/ "&/3PQR-!&$4:U234''/:52:S 0!8+O!&!:^0M3. @ DR-!&7& " @ U^0& 0_3^ "&Q "34:P07'! RS!&0'D ->RS!3> q? @ J0T'J:0 ')(*4: "!'51U|3^!03!0!c'i "k "U.'GPS'G(*!&Z3]!03!Z0(!^MQ [Yv^#d 0c3Z0 "&4^
z(x) =
0.25(cos(10π(x− 0.5) + 1)3
1.4 ≤ x ≤ 1.6
03/ "
v "&0/ "TN'i "U=0 "&4=
u(x, 0) = 0.0 et h(x, 0) =
1.+ ∆h− z(x)
1.1 ≤ x ≤ 1.2
1.− z(x)3/ "
.
4L∆h
3='/7!&!0^'G(/51!' "PS3!^ "#%T' .V=Q "!&T'/0:!8+c)Y(a)
_.0&_/51!'/3/ "rP8(*:33?gq?0&/∆h = 0.2
Y(b)
x\..^/51!'/ P8(*:3V?gq?X0/3∆h = 0.001
YN'51U`P&vXW;RS!fB343 "'/!c^3 "M' A 51^C9 g = 1m/s2 \J68,+ b\=!+1`9":g = 9, 81m/s2 Y
'"
E*9 #%% %
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4le fondla surface libre
v_#d "0c:='N73!3#%]'//M&_g7 >vO4:!'LS!514RS!L M&.:-!&OL'&452 6&<ZHZI 6 E 9" "e @ !& ^1Y n&P$ZYCUK]3 "U> 52.M'fgL:!8+ #d "!&Df! &452L4RS!/'/M3b< -\?;H . \52:/''/!&$RS!=!-+b M&.:-!&~'/3452D< 8,?;H. C9 !&2.51U;'/3/RS!=0!.:51]3 "!31mW_6&(-qpY
l l k7 % 9 gp gl k K Kk k # En6n6n t k :P "!''/ "Y.:3.:'G(651`:4$0!^4:5bZ6&<=HJI 6 3!!^ "M&' A 51Tm*68 @ (-qp)P 4LD'"]=0&k: ":g#d "0]~`9". "Y @ $ "f4 @ !'/ A P z(x) = 1m
3(25/3)m <
x < (25/2)m:z(x) = 0
'/'!3PrUDR-!&b'N' " @ !:!_. .'/B011M3V0
25mY N'51U7'/L!V#%c'/Mb32g
10m\2'/LRS!U..40c51 "!&9":52:U2:3
−350m2/sx < (50/3)m
\350m2/s
" P:!&. "!&b''/ "&> "3/04::b'/:4:!'D02'Ne3!3#%F'/M3E @ !3 ZYK^\7'204:MEd @ !: ZY -"K^!8+/VU
0sP0.05s
P0.45s
:0.65s
Y &J'/:k ":PY 4L'N>. "Y @ _3Z4 @ !' A 3^'N "'!&. "cVJ: "52Y "34]0f0&!8+ "&0Z0f4\#%:/ "`9":D0:Zk: ":JgB#d "0fU.3:'/'/_::Y$9;`Vx fT51/v:]4\9-/0:'G(651`4$0!]4:5bZ6&<=HJI 6 Y "!v'/|/5D!'N/ "S!524:R-!&_0_ M' A 52Z0]6&SV?XW;:U`9":^ "Y @ ./Z/34 @ !'/ A Z0T' `T 4L/'r13. "0:=k: ":Z A :)YHZ "34!&'q. "U= 52.M'_g>:!8+01m 6- @ (-p:Y
'
%E * %% 2 *9**2 * *
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.9999
1
1.0001
1.0002
1.0003
1.0004
1.0005
, B"F<k -] &(G-J/21:+V'J143<5"M+\:M3K33 > K(G)8'P"
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
!#"$&%' (*)+-,/.1032 .465 7 280 434*4:9<; =>?
0 5 10 15 20 250
5
10
15t=0st=0.05st=0.25st=0.45st=0.65s
@ A8BDCE(FG)IHGJA( + K%LM6HG+N,/.4I5O$2/0P4*4*419<; =>?
0 5 10 15 20 25−500
−400
−300
−200
−100
0
100
200
300
400
t=0st=0.05st=0.25st=0.45st=0.65s
Q A6RFSCTU"$V MIW"$X%Y' P(Z)+-,T.4I5O$280 434*4:9<; =>
[8[
E*9 #%% %
jOkgMk[ l k kk N 9n 9HZ "!&$ "!$30& ";0:;`3C90 "!&M'/|3/ @ !'N34Jm $ h|8pGP!T @ !'/.4~!9`!O0&_ 0/ "&J/&./'/:T'/!& 'N10/3 "U.S!4D0!L#d "&0Pr0 '/: ` 4::S4'/ #d "&0e:3=:4:U4DT'/D#d "\./ K
z(x) =
03
x ≤ x0
13
x > x0
:='N7: "0. "/./'/=VJ0 4^
U(x, 0) =
Ul
x ≤ x0
Ur
3x > x0.
^gO'NL3 "'/!8./ .4: "R-!&'/:!'/4: $Z'/3!0 a:2h~:,+S'0 "! m3$Jh|&\pGP$ !'/' "i4:9'/!&T'G(3:!|Y "!T'Nf "51
L1 0_^&452B:e3343UU='G(* 51 @ _0Log(‖u − uh‖L1)
P 4Luh
3]'/F "'!&. "OS!524:R-!&P:#d "&:. "O0Log(∆x)
4L∆xV~'T;0([Y&~ 5251T'N 34:/ "B0!14:5b]33! @ 41V./:!'/$!1&9"!0D4:q.] "VU7 "!D'/' "] 51:D'N "'!&. " '-/RS!B:DS!524:R-!&P0b'/
!V#%]'/MP&'/7R-!U.4^0^52 "!89"51U)YvF "57M&b0& "!0&bVB04:D
F =u
c
Px 4LuV>'Nc9-.:30! !0b\
c =√gh
:3^'/b4:'/434PrRS!$:3^')( '/ @ !f0&> "57M3>0 "aY "! ' @ k>08852RS! "5133/M'#Y68 |F | > 1'G(4: "!'51U_V^0 !Y3.R-!&Pr0_'/D`_ 4L |F | < 1'G(4: "!'/:52:U:30&x!MY3.RS!#Y x'/`x "!7')(*4: "!'51Ux3|0!f`3!MY.R-!&
!`!&Q:3/RS!_ "!c9-/\9":.1'N73.!. "VZ0&._.33.RS!#Y
nn) k7 % 2HZ "!&^'/' "= "5152:: Z.V.:_>&452F!_!a 52&'/PYL#%/.S 'Nb: "5B?/3 "0b'N9-.3F0&b S9 @ :c0! &452a6&<ZHZI 6`9 0/ "&]N'E%0^`
x0 = 10K/Y
hl = 5m hr = 1m
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0
7
4"+ Z[(%!#"$.-(0D8@J.*IC2?"$!% 2-f"$8; O[(0!0"8"$!% Cg A"J[ i 2O k 6 Q',qpH. T2(%7.YS[ !06 [HO.',C_\[ !%'.'(3_\[ !%''qpE76D('!%,6(')"!% ` 6 .8!%W6M K38"d +(0!%E- 8f2l"$!% YJ-K3^c7(%!0W36CV F Z[(%!0!0"E('9c!% +C6*/:6[ZYJ-K3!%W6[YA8(0('C_ !0(%(0Y" Y (%(06 ?"9 4" .(06 (0&c(%d& /:63Z c!#T$/:6!%CX: 6 !%7?T(%!0 Y .'.pD;2 (%S@J.Y U P3` f"$86!0"176(' !096(%)"$!0 c(%d JCgd ,:) 9(% ;)XC6 3d(% 2Y23!#"$8XCX.8!02#Y 9(0Y63d(%Dc!#T$/:6!%CDC(%68!%6C+@JC@JC68"(0+W6SIH6 !\_ 569c_7P)"$D"5OC :63T(%!06 P3.'0_X 6 Y + . 3_ I d.+([_X FH.'.p=_ 4" 6 68Yh`(\<]?Z3" 8!% b8@J.2(064% !%b. 4".Y.8!0" 76 (%bcC(%d b ./6!%b "6: 4"X(%^8"6c:l"$!% A6 K38"d , %986C"$ 4"++82[!08"6(0(%I6+88!% C+MeT"$!0
(p− pik)∆αk
_k = l, v
Y!6K X85c:$) "V6aV"$;2)6[Z F8" VIR.'0p ;25(0('c7)"$6I 4"-6C"$!0(%!0.56 D ."$JC[DD^8"6c:l"$!% A:)C 8!0"$.D76-8[@J('9c!% 6 /:6[Z:6 )"$8!%IC!'g :(%!0$cC(%F?"E!%(%K "FC(%!%W6.I(0@J.YDP3^6CE(%D8!%96C(')"!% N6c(%d-P-) Y CF- 4"$?Z3"_ Y J!0:g(0!'"M,6C"$6D 4"M78.6 L@J.Y1b2(06 %: !%D. 4".Yc.6D(% /:6[Z;".8!%8"!%W6VC.8 4"@) "E6 CIY)"!0F8!%g CV6Ng!% 4"K6 /:6[Z9J-K3!%W6 Y 9[\ .'(p!%(0H(=<> "C(%!0W36C.E76H('!096(%)"$!0 5H.6(0 4"$!%J!0W36CH C[!0 !0 (%",96(0"!%!0 4"$!% (0 Y 9[\ _ F6!.'0qpY 6K 8"$ 4",(\<> :( K3 <>6C @J.YA2(06 %: !076(0 K38"d - 1J gd C f"$86!0":LF >=J:(06 a )pc:8. 6Y(\<>6C"$!%,6('8!0".&(%L(%6C"!% 6 c(%d1`P!%Y C J gd C7(0.&OCPQ+R W36C!a Y;(06(0W6X(0 /6[Z CJK38!%W6_ C6 C8"$86!%8Y6 6C2;6@J.YC. 4"$.IC.8 4"@) " 6 V)"$8!%a8!%g a: (\<]."$^a.!%?"$6G76. 4"$8(%."$)"$6`(0!%6 9. "D(0!/6[Z^_ 9: P[50_ Y 9[\ .'(qpY F('X 6^ !0"I <>c["$ C!%8.6C(0"@l"$I"$8d 6g;) "-76I(08c(%dIJ-K[78c^(%!%W6J gd "- 1J gd Mi /*!0"D ('N8 !%d8:8"!%;k/YCU6AW36CM(0c(%dc!#T$/:6!0M X "9:9J-K3^c7(%!0W6_ 6 (0!%6 X(%6(%,(0Y!0g X('1Yl"$!0 6g$)!% "+6 /:63Z _^ (%6(% 9868"9(%D!0g , <>6 ` 42"$65W6!G78?",. 4"$8(\<]."$)":) (\<]."$7,9C.C!%"6+" 6C!0"$, `6C"$!0(%!0?"5."$)";2N(% /6[Z CJK38!%W6: D(\<]."$7N8?"$6)Y F(%& 6 C 6C L8@J.1C. 4"$. N765DlTc(%d (% Z[)YH) M@J!0"b 6C,(0(% C.,% !%(%b [Cd(%NC!%J:)!%W6b:) ('N8 !%d85"$!0 _ ":) ('N63ZC!0d ?"$!0 A C6-(%(0 V.?2(%7('N ."J[C6C"$!0(%!0. ^"Y^6C(01Y <>6 K38"d $ CY(\<>%986["C('S88!% !% 4"$8/*!'(0_6D6 L;4LL(%52)(068DC8D 4",h18"!%N!%g!0 :!0M/*)!%c(0i*MC6L8c(0d : P 8 ;2
αv,init = 0.6k/Y ; 6C"!%(0!%(% ."J[CA Y+(%6g W6!^ "
,(%6(%!%8" 4"(% )"!%!%g Mc.96!0".$l"$!% M(=< )(%g8!0"J+Q-&'I"$ [TeO[@J6(0"H4 -(%-.'0p\_C"76I(%+;Mi
αv,init = 0.2ka6(%V2)(%6I88 4"6
:f"$!%D!%g!0 :!0+C.7 C.$) "_7 %986C",(%bC8!0 !% 4"8/*)!')(%_C"- 6C"!%(0!%(%.?"$J39C .Y1IH6 !G"5(%(%c7$)"68 F 3_ VEIR.'0_1I 9F].'._1I 9.'([_1IH6).'(pY 6K % 6,(%(0 D(0!%W6M(08@J.SOCPQ+RO ^6C,('!096(%)"$!0 S 36C.!%W66c(%d5DP 8 Y
0'0
7 ! (
1D
(*) 143*6 ! 6,3
b"$;2)!%(a 6b !0.8 N6 [d(%XcC!#T /6!%Y.g!V:NW6:)"&.W36)"$!0 h8!% 96 )_4 4"a6[Z.8!02 4"V('5 82))"!% -(%5Y8E6[ZJ:8"M63Z .W6:)"$!0 D.!#2 4",('& Cf2))"$!0 6L "DMC6[ZLJ:)G+ ,;(05 K38"d ,;2M !#"$!0 &!% !#"$!%(%<].!0"(%9Y C!%d856!#2) 4"$
∂W (x, t)
∂t+∂F (W (x, t))
∂x+ S1(x,W ) = S2(x,W )
W (x, 0) = W0(x)
i * k
;2
W (x, t) =
αvρv
αvρvuv
αlρl
αlρlul
, F (W (x, t)) =
αvρvuv
αvρvu2v
αlρlul
αlρlu2l
S1(x,W ) =
0
αv∂p
∂x+ δ(p− pi
v)∂αv
∂x
0
αl∂p
∂x+ δ(p− pi
l)∂αl
∂x
?"S2(x,W ) =
0αvρvg
0αlρlg
;2v"l8$)^8"$ 4"Y)6[Z J:8g4+46Y?"Y(0!%W6!%CN7?"$!02 4";_
g8"
(\<>.(0.$l"$!% `b('X7$) "6_αv
_ρv
?"uv
"D87?"$!#2 4"M(%9"$6[Z`Cb23!0_ (% C!0".F?"G(%23!0"8FCF('I2)76_
αl
_ρl
"ul
8 4"G88^"$!#2 " (%K"@63Z,Ca8.8 _(' 8!0"."X('`2[!#"$`C6 (%!0W36C!% ;5 C"&!% C 6
(αv, αl, ρv, ρl, uv, ul, p)"
W6:)"$8L.W6:)"$!0 _I S!%(5 C6&/*6C"1A(0!%X <>.?"@)"A76X/ S K3f"$d !E[B!0d8M(0!B(%!%(0"@6[ZAD23!%"
αv + αl = 1_^?"('b6[Z[!%d ,(0(%+g4+4D:8/*)!0";_
g6C28 A(% 8!% / ?"$!0 b !#"$. C1(%A2)76b"bC6 (0!%W6!%Cp = cργ
v"ρl = klp
a $# c_γ_a?"kl
8 4"Y f"@ 4"$Y .c = 105 _ γ = 1.4
_a = 4.37 × 10−5 ?" kl = 987, 57
"; 678b(06W6(%& 8!0". C6L(0!%W6!%CN8"W6:! T 8"$ 4"$
ρl ' ρ0l
"aW6+(%DCJ:2)^6F8"a"$da(0.gd:E$C^f"VhD('J:8(%!0W36C!% pi
k
!#k.8!%g C
v"lf"K('+C8!0 N!% 4"$8/*!'(0aC 6[ZNCJ:_(0a" V! . "!%(
δ(p− pi
k
) ∂αk
∂x
8" (%Y"8&88!% 8"!% %'& (OK `(06!06892)!%.?"$.N(\<]?Z[88!% (
p− pik
) P-RE0'0p=_ FH.'p(& ! 6C"$ I(',(0! C/ "6+ /:6[Zc6(0(%IC.8 4"$.M:P) I* [BJDK [BJ).qp W6!B.,% !0"
p− pik
86!0"
p− pi
v = 0
p− pil = Cp(αv)ρl(uv − ul)
2
;2Cp(αv) = αv
6Cp(αv) =
αvαlρv
αvρl + αlρl
_:;2δ8"6 CD f"@ 4"$D@J!0!0!
('986!0" ^8θl = δ
(p− pi
l
) +-,/.1024365,
0'.
! 7 ( !
Q IW! !"$#%&''() * !&'&+(,- .# /",/0#1/ "324# ",(650#1()* 798:;<#<=>?@A=*#CB D"$#EF:+8G HI;&'&J.#1()*.# & ;(6:D(,")()':!.#1&& IK(L+E# "( M&HON PEIR.'0N IQ9F].'.N IQ9.'(N\IR!.'(p(QFS#1&T &U",&H V"W24()",(65#1() X")#:;8G Y [Z ")\ ")-.+0pWNC() ]&[")Y^Y KG_][#0B
δ = 0
(*)_` a "5 026+b*."6&1Dc 3 ;d # .Ke*143
PA# "$#F&()];&H.# &_()H DT�B1# ()"fT# .# " KGTF#18:F#1(,gD h98:;<#V=>?@ L)ji!I &V");V KG_]U8-KGk&Jk",()gUl8\];U#m#1JIO(AnolB 0# 98:;<## k",:D=>?@A=V :&%.(fA(6
W ni+ 1
2
=1
2
(W n
i +W ni+1
)− 1
2sgn
(∇F (W )
) (W n
i+1 −W ni
)
+12R(W )|Λ|−1(W )R−1(W )Sn
i+ 1
2
W n+1i = W n
i − r(F(W n
i+1
)− F (W n
i ))
+ ∆tSni
p * q
#0B ∇F (W );'r")!\ &# (,s /.Ot-8KG'()g
F (W ) :,%.(Rnu"W24:+9#1 H>O
Λ(W )'r")#EF#1&();A ()# \.#1")'%J"$# J",# -\ (); /.Gt-8-KG()g
F (W )
R(W );'r"$#%F#1&'();A B;;&'&'&;; R−1(W )
'7(,sB&' PA# r")!# $#-"$#%F#1&'(); ∇F (W )
;' ($# \ .# ",()#1J")
sgn(∇F (W )
)= R(W ) sgn
(Λ(W))
R−1(W),
;r =
∆t
∆x
#CB ∆t
_r")!.# !;∆x
_r")!.# v2w'.# ;!
Sni+ 1
2
;'H"324# &0tO()F#1(,h x;&'&'s&'[tn, tn + θ[×[xi, xi+1[
# y(,B# z "32w:;# IS&:; ()+&;
Sni
_f"324# &'0t(,<#|() !;& &;[;s&[tn, tn+1[×[xi− 1
2
, xi+ 1
2
[ .# "W24:+9# k!&'&;;;& [:;#1 H>Ok&/&'J",]!J( /.() A's :,%.(, .# IQ9F].'.1i
1
αl
=1
2
(1
(αl)L
+1
(αl)R
), αv = 1 − αl,
p =(αlp)L + (αlp)R
(αl)R + (αl)R
,
uv =(√αvρvuv)L
+ (√αvρvuv)R
(√αvρv)L
+ (√αvρv)R
,
ul =(√αlρlul)L
+ (√αlρlul)R
(√αlρl)L
+ (√αlρl)R
,
;
ρv =( pc
) 1
γ ;ρl = klp
a
.'
AX+ 7 v ! m (
1D
$#(.)L
;'"32w:;9#|/\s# 98D+(.)R
_"W24:+9#1 &(6 FY()'gGA") KG_]AJ( /.(, v24;'/# L8-KGk&Jk",()g NO# ",& ∇F (W )
v2w_.# L ($#1\.#|5")(,# J", N !()"k # O/U B +&rk&M :;s&'&"32w:;# I&:; ()+&Nk&M")#*:&'(,",! KG'] p * CqM")# &!(6B1# s
∂W (x, t)
∂t+ ∇F (W (x, t))
∂W (x, t)
∂x+ C(W )
∂W (x, t)
∂x= S2(x,W )
p * ,sq
#0BC(W )
∂W (x, t)
∂x= S1(x,W )
6Y(6U ()",()")<98:;<# &# ;()s < Oty:;# k KPA#1T")#*&'()];&E:;# Iy(,")()'u")E98:F# v2 6X")&H v2w()s:;\&#|() +tO",()(6!;*
∂W
∂t= S2(W )
W (x, tn) = W n(x);#0B;D",!98:;<#F=>r?@A=U-&:;O",! KG']
∂W (x, t)
∂t+∂F (W (x, t))
∂x+ S1(x,W ) = 0
W (x, tn) = W n+1(x)
p * *sq
PA# r# "$#%&'()]&'!:g.#1()- * KG_] p * *qM2w:;&(6
∂W (x, t)
∂t+ ∇F (W (x, t))
∂W (x, t)
∂x+ C(W )
∂W (x, t)
∂x= 0
⇐⇒ ∂W (x, t)
∂t+ (∇F (W (x, t)) + C(W ))
∂W (x, t)
∂x= 0,
*I' A(W ) = ∇F (W (x, t)) + C(W )
[Y24:+9# kD&: ();&r&+BG()sAnF# &G98&r"$#E ")()-+t# +D &'J",]H T>();<# (6B1# s
∂W (x, t)
∂t+ A(W )
∂W (x, t)
∂x= 0
W (x, 0) = W0(x).
p * V q
PA# "W24:+9# ko&': (,;&;N (,:\&# s "32w:g.#|() p * Vq* .# ", F# (,[tn, tn +
θ[×[xi, xi+1[ 6Y \s# &' .# su");uF^;F;% #1(,%g-;")",% *")#&'()]&'U.# &_()<
# &#
W ni+ 1
2
=1
2
(W n
i +W ni+1
)−
αni+ 1
2
2Sni+ 1
2
A(W )(W n
i+1 −W ni
).
p *Z(q
6Y(6!*# ")(,gGA"$#T:;8G A !k&_&'J.#1(,U.# &/ '(,:!&k:;!.# &Z M&_;N 4PJ'98+& IK( PIR.'0N M & .'.1i I+.# s")<98.# \;F;sE VB1# &()# J")(6B1# s
ρv =ρv
ρ0v
+ρl =
ρl
ρ0l
#0B ρ0
v
;ρ0
l
sr Gt* '(,:r0# &# ;:&(,'()g&'k 5
(6BsM A"$#!B1# k&L; U")()g(, Ns;(,< :&%.(6ε =
ρ0v
ρ0l
! MV#DI :
.
! 7 ( !
g-"$#8# VB1# k&u;'u&]u",:\];&V# &%I &'Enh")#8.# '-")()g(, NR# ",&ε << 1
PA# r#
A(W ) =
0 1 0 0−u2
v + αvp,1 2uv αvp,3 00 0 0 1
εαlp,1 + θlαl,1 0 −u2l + εαlp,3 + θlαl,3 2ul
.
p * q
6Y ;.# s<;* # ");" # (,E # M!.'0ON Y Qd[\ +ir .# F")-0# $# ") /.() ")(,gG() D'(, F&''()J") Nkx#
αvp,1 =γp
ρv
Np,3 =
γp
αvρl
NFρl
;'!;'# sH
I ρl = ρ0
l
*# &#p,3 =
γp
αvρ0v
ρ0v
ρ0l
= εγp
αvρ0v
Nαl,1 = 0
;αl,3 =
1
ρl
;A7k
γp
ρv
= c21+
c22 =θl
ρl
,
;r-&;F"$# ;! .# p * q N7# &# A(W ) = A0(W ) + εH(W )#CB
A0(W ) =
0 1 0 0−u2
v + c21 2uv 0 00 0 0 10 0 −u2
l + c22 2ul
;
H(W ) =
0 0 0 00 0 γp
ρ0v
0
0 0 0 0αlp,1 0 αlp,3 0
.
[Hk" KOF0# &# ;:&(,'()gH# 'O;():n A0
'r :.# &
P0(λ) =((λ− uv)
2 − c21) (
(λ− ul)2 − c22
)
k" KG# ;g.#|&!B1# ",&'/&& (,'()+g('s
λ1 = uv − c1, λ2 = uv + c1, λ3 = ul − c2+
λ4 = ul + c2;
R(W ) =
1 1 0 0λ1 λ2 0 00 0 1 10 0 λ3 λ4
.
6Y7&'"$## s A(W ) = A0(W ) + εH(W )7# &#
W ni+ 1
2
=1
2
(W n
i +W ni+1
)−
αni+ 1
2
2Sni+ 1
2
(A0(W ) + εH(W )
) (W n
i+1 −W ni
)
=1
2
(W n
i +W ni+1
)−
αni+ 1
2
2Sni+ 1
2
A0(W )(W n
i+1 −W ni
)−
αni+ 1
2
2Sni+ 1
2
εH(W )(W n
i+1 −W ni
).
.
AX+ 7 v ! m (
1D
MFε << 1
NI")#8:&'()H Tk&_&'J.#1(, Ak:&")():0# (,& @(,). N1 M8.#'0+0ONQ#1 .-VCiS&B Tg%'(R I+TTk&_&'J.#1()7I&I:;&")(,:0# (,& B :;&(,B1# s B′ = B + εH !#
ε'F &:;"/&']I+(,E+ B ;'F ($# \ .# ",()#1J")F .#
R#0B ;B1# ");&&&'/ (,'()+N# ") & B′ _ ($# \ .# ",()#1J")/'&R#0B; B1# ",&'
& &;M (,'(,; IQdF].'.|iI+M",XB1# ");&X&'&;λ′j
B′ sM# &'O98:;L.# &[","); B1# ");&/&&'
λj
B .# &"$#%&'"$#|() U(6B #1
|λ′j − λj| = (ε).6YV# s F ;&:;"69#1MVkO (,&rgGA"32wk:&#|& A ;'/ ($# \ .# ")(,# J")&
R+Eg.#1&UB1# ");&u&'&'Es (,'(,;;NY()'gv24;")",u E&]E&'O98
T;",") A0
;A # ;%0#1hkO&# (,&")D KG']%.# &r")H(,\D T")#FF#1&'()A0
# ")(,h '()\u %"$#VF#1&(); A ZS().#1")s")%'98:F#*=>r?@A=;24:;&'(,k&A",& J")];F!J( 5 /.(, p * jqM "$#%F# (,]&!'(,B1# s
W n
i+ 1
2
=1
2
(W n
i +W ni+1
)− 1
2sgn
(A0(W )
) (W n
i+1 −W ni
)
W n+1i = W n
i − r(F(W n
i+ 1
2
)− F
(W n
i− 1
2
))+ ∆t(S1)
n
i ,
p *Z0q
#0B
(S1)n
i =
0
(αv)ni
2∆x
(pn
i+1 − pni−1
)
0
(αl)n
i
2∆x
(pn
i+1 − pni−1
)− 1
2∆xδ(ρl)
n
i (αv)n
i ((ul)n
i − (uv)n
i )2 (
(αl)n
i+1 − (αl)n
i−1
)
.
(*) ( .c 6&1Tc 3 ! 5 e # d b 1 ! 0 . # *6 ! 6,3 c1Tc # .026 d)
[Y2 # ")(,0#1()z m98:F#x=>?@A=yk&D")F&J")]J( /.() p * jqH # T;V98.# (,&'# :+:zF;:x.# &U"W2 #%9' - &'Fx x"$#m&''() z &&;;() ;7"$#o:;8G x I;&'&J.#|() *# & '(,: IQ9F].'.N: PIR.'0N: M & .'.jifI&G98;&r")#9# J(); \ ();# &-sB;;& A0
()# \# ")(,# J",!PA# !;;'u;;() !# ",")r (, :&'&A",% KG'] p * jq I # s
δ = 0 .# A",
;&u ( I:&'s(,"δ(p− pi
k
) ∂αk
∂x
;# !&' &'H#V")#VF:+8G % uk&_&'J.#1() .#1& (,:/E0#1")"$# s")#A<#|&(,L()\ ()s&'B .#1[ .#1Y"$#8.# /&: ();&Y '98:F#=>r?@A=U*()")(,# s/")#%F:+8G ! HZ ")\ ",-.'0NPP + .+.1i [f#!9#1J(,H s *&k # ");");&/")!(,\! .#10# ;' :;.# &
A0
(W)
=
0 1 0 0−u2
v + γp
ρv2uv
γp
ρl0
0 0 0 1αl
αv
γp
ρv0 −u2
l + αl
αv
γp
ρl2ul
.
.',
F 0 v U f 7( 7A 0
PA# ")!# !#");/B #1")&& &; A0
(W) srr HFG ",!")k;(,rg
1"324# ")\ &(,8r D?r,'/O5_=G98G" 4H'r :.# &
B0 = A0
Bn+1 =3An − A3
n
2.
;A&Y")YB1# ",&&&'[ A0
s[.# Y ;[ /FG ",")[k;(,Yg1N ;[
(M > 0
*#
sgn (A0) = |A0|A0−1 = |MA0| (MA0)
−1 = sgn (MA0) ,
# ") &[<%",(,")(,"$#F#1&'() A0
(W) .# &
M r",") &'g",LB1# ",&'L&&'L
MA0
(W) 's DO ")",k;(,rg
1;# ",()gH"$#uF:+8G H %Z
",\;/I&MA0
FR&r;&J")]!*&;
M =1
L0
#0B
L0 = max
(|uv| +
∣∣∣∣√γp
ρv
∣∣∣∣ , |ul| +∣∣∣∣√γp
ρv
∣∣∣∣).
=O(/UB1# ",&u&'&'U'BG&# (,F;s%&']EI+(6 NS#xsB;&\;7;&#x")s!XPA# ;0# ()"M'Fk'()J",* 2 # ;:;"):&'&E"$# &'O;: &!M=OI'g7",uB1# ");&E&'&' A0
(W) (,s .# H"W24(,s&_B1# ")",
[−L0,−1] ∪ 0 ∪ [1, L0]p ;(X_I'(,J")E# &]")#
%"6()")()#1(,U A0
(W) .# &u'9#1q+Nk k
a0 =1
L0(L0 + 1)
NB0 = A0
;r"324# ",\&(68r2w:&'(,
Bn+1 = Pan(Bn),
Ln+1 =2(an + 1)
3
√an + 1
3,
an+1 =1
Ln+1(Ln+1 + 1),
$#Pa
;'r +(,-I" KG($# ", :&%.(,H# &
Pa(X) = −aX3 + (a + 1)X.
FR&r") :+9# (,"v&"W2 # ")(,0#1()F ;':;8O AB ()& ",-.'0Ci ? Xgr;'rF:+8G ")X^X&:;"69#1XGF:;&()g;Lgr"$#A:;8G "$# :;Fk'(,(,*k;&# ",T 0# ","K T")#FF#1&'();H'()\T .# A")H0# $#
δ 6= 0p B(,&
%.\&' Q* ON ]* .sqD+#h"324#0B #1# \V UI;&;u")#h()T"$#1()z:&(,gU & J")];FA ># .# /",A0#1/$#
δ = 0p B(,&:%.\&' )* N )* ON )*Z,ON
:* *q
.+*
AX+ 7 v ! m (
1D
(*) b-b &6 5 d ."6&1Dc !*# 026 ! 1 *6;c .3 dHc 1 #
& "X24# \(,H v2wo# _TG (,F;(,;"()_9#1().# ()&'%&'I :<# & 4@ K>#1Jk98<# & =!:&(,g!I &/");/G !8&8G & ")()g ()8# (,gG
1D >r@ 1i ;A (, ]&'T"32w:;",sr v24T;") v24# nE"$#E&_(,H 24&J(,;r :;JI 98.# s .# [;;()ss;.# sL "324# ()&f#BG(6'r v2ws&': r"W24# '[
10 m/s# ")&'
gD"W2 #1()&M;'# -&;I &# k&_ ;;() ")#%J "W24;;()s_"g",9# Gt< BG() r()s:;\&:&L"$#H+(,<B1#
0.2OPA# L;'A; %.\ ()NG VJ'&'B A
8:;]u T'&'()+() h 9';A!"W24+A u")#Fk# s& YxI;N(S # (6A"32w8s5I18];HgGH",9';&''D;8:;&s p .#1/ v2 #1&Æ.# \ ")(,g() &'FA H\ ;N.# r HI:;:;()* "324# ()&/ # r")9';9q+N "324# ;:",:(,U ",()g(, ! Dn")#uI;# s;&s&#$<:;'# (,&;F;%o&:+&':(,'s% F"$#*+(, F.# '# \< o")(,g() E.#1&&_B1#1(,V F :J(6 OP");NO",[ (,(,[()(6()# ");&'&;k .# s/nT")#H'")O() gv24 JO(); &# (,T "324# J;U *k# s;& p
αv,init = 0.2NR(6
αv,init = 0.6.# &_9q+N A# Ot- !BG() !& .# \! ")#%+(,- v24;s&:;HB &'r")#%;;(, &_(,n.#1&'(,& "32w()'9#1()(6($#1" (,()#1Ot-")(,F(6;M r&'\&# F:; "$# #;-(6B1# snU"32ws&':Ex()I %'&")AB1# &'($# J")
αv(0, t) = 0.2Nuv(0, t) = 0
Nul(0, t) = 10m/s#0BD&_(,!")()J&!S# ()/gHI &"$#u&;'()**()I'AH &'(,p(12, t) = 105 #
#0BHs&:H")(,J& u0# ","R ()&';A (,\% u")# # ;J();%# &A"$#FF:+8G % EZ 1")\ !#0B;E"324# "65\&'(,8Fh y?r,'/O5_=G98G", ",|iAI;&;-"324# ")()#1(, '98:F# =O>?@A= # & J")];FU8GI;&Jk")(,gGAn% (,()FgA"$#D.# &'()()F# \()# ()&'/ MB1# ",&'&'&;(6Sk;(,[+k&'F+S")#r (,F(,(,! M")#r ( I(,D:&'()gL()s&'O (,Y'&S",L'98:F#p B ()& %.\& N q)Yh&';B1# 98T"$#u:;8O H &K(KJ# :;D'&")#I;&'&J.#|() h '(,:7I;&;<"324# ",()0#|() 98:;<# =O>?@A= # &J")]* >#1 # L&_&'();()&L")#B1# ",&X 9# Gt rBG(, ")#B1# k&[ .# L")#H; (6() ()(6()# ")%N ;F")0#1.# &r+tO;F",
αv = 0.2p B(,& %\&; "!; q/g(
v2w'.# k'()J",!#0B;δ = 0
PA# ["); %\&; #"!ON $"%ON # ON #"N &'N #)(ON #*N %u"$#uF#1&'();A(,\ .# r"32w:;# k!&:; ()+&;'0# ",",:!.# &")#E:;8G )&R F( "+ &(N + P ,NQ NQ-%GNK%1iJ.# ':/&"$#k&_&'J.#1()D.# & ;(6: # &Ls&")#HF#1&();/(,\r .# L") %.\& &*N & ON #".ON &/T_0# ",",:!.#1&r")#%F:+8G ! HZ ") \ ",N10P + 1i
!
X+ 7
0 2 4 6 8 10 120.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.7648 points100 points200 points600 pointsanalytique
! "$#&%')(*%&+,(.-0/1324+5%62374138#9(*138:8;/<+,')8>=<?@8;+BAC0CEDF-0(./<+,GIH
δ = 0H
J4K +,LM-3138N138NOEPRQS.-02MT8UG
V
0 2 4 6 8 10 1210
11
12
13
14
15
1648 points100 points200 points600 pointsanalytique
W R " #&%6',(*%&+)(X-/ 1M8SX% #9(Y+,8IG,GZ8 1M2 SX(X[<2M(*138 8I/<+)',8=0?\8]+^A0CC\D_-(X/0+5G;H
δ = 0H
J4K +)LR-91M8N138O@PFQ`SX-2MT08IG
0 2 4 6 8 10 120.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74delta=0delta=5.10−4
delta=5.10−2
delta=0.5
aW.bc"ed/3fR2M8I/Rg]8h132iDF%&j'5% J4k +,')81ml L9n<D_8;'M_-SX(Xg;(Y+ K
δGZ2R'SX8
+5%6237o138p#9(*138αv,init = 0.6
H@qCCD_-(X/<+,GIH J4K +)LR-91M8N138O@PFQ`SX-2MT08IG
0 2 4 6 8 10 1210
11
12
13
14
15
16delta=0delta=5.10−4
delta=5.10−2
delta=0.5
.h"rd/3fR2M8I/Rg]8p1M2aDR%',%6jJ4k +,')81l L9n9DF8I'GM_-SX(*g](.+ K
δG)2M'sS*%#9(.j
+)8UG)G)8@1M2tSX(X[<2M(*138αv,init = 0.6
H&q6C0CD_-(X/0+5G;H J4K +)LM-3138N138NOEPRQS.-02MT8UG
uwv
N XN+ v mN4
1D
0 2 4 6 8 10 120.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5t=0.1 sect=0.2 sect=0.3 sect=0.4 sect=0.5 sect=0.6 sec
W "%6237 138#9(X1381M8SX%#&%6D_8;2R'
δ = 5 × 10−4H150
D_-(X/<+,GIHJ4K +,LM-3138N138d PB-2 J (
0 2 4 6 8 10 1210
11
12
13
14
15
16t=0.1 sect=0.2 sect=0.3 sect=0.4 sect=0.5 sect=0.6 sec
cW v " (Y+,8IG,GZ8 132 SX(Yj[<2M(X1M8
δ = 5 × 10−4H150
D_-(X/<+,G HJ4K +)LR-91M8N138 d P -02 J (
0 2 4 6 8 10 120.15
0.2
0.25
0.3
0.35
0.4
0.45
0.548 points150 points600 pointsanalyque
W V "s-02M'MF81384+5%6237 138#9(*138138 S*%N#&%6D_8;2M'E8;/<+,')8`=<?ND_-(X/<+,G8;+ACC D_-(X/<+,G
δ = 5 × 10−4H
J4K +,LM-3138N138d PB-2 J (
−2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
Log(dx)
Lo
g(L
1−
err
eu
r)
y=0.48
" s-2R'GM_8 1ml 8;')j',8;2M' 8;/ /M-' J 8
L1%0G)G)-3g]( K 8 %62
+,%237 1M8e#9(X1380H D_8;/<+)8' 0.5
HJ4K +,LM-3138138d PB-2 J (
u V
X+ 7
0 2 4 6 8 10 129.7
9.75
9.8
9.85
9.9
9.95
10x 10
4
"
δ = 5 × 10−4
150!" #$
%& ')()*(),+-./ %
0 2 4 6 8 10 12−25
−20
−15
−10
−5
0
5
0132547698;:)<>=@?BADCEFFEHGEHIJLKMJONP ERQSTUCWVXY ZHFE\[
δ = 5 × 10−4 ]
150P_^ A`"CaF ]bc Cd ^ G)E,G)E,eYf ^ Q b A
0 2 4 6 8 10 12988.0658
988.066
988.0662
988.0664
988.0666
988.0668
988.067
988.0672
013254g698;:h:i=kjBER`FA#C c GQlI#A#Nm QAnG)E5ToCWVXY ZpFE[
δ = 5 × 10−4 ]
150P_^ A`$CF
qsrutvxw9yz9Uz9H| 8s yh~9r
0 2 4 6 8 10 120.98
0.985
0.99
0.995
1
1.005
013254,698;:=gjE`FADC c GEpInJKMJONP ERQS,TCV@XY ZFE[
δ = 5 × 10−4 ]
150P_^ A`"CaF ]bc Cd ^ G)E,G)E,eYf ^ Q b A
h
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")1i Z ")\ N^]_D)3S*,-/;(<C+-P`)"a(?(+3!C+b.C+3dc+>@(C#5ObU7(+*Be+$'Gf-SgBM7(C+N#R;986&;I &'0NO&# G5k&_ + N)
0L(ihji Z 0M;j8.# ", vNi #"ED)GfbC+-VCYD)"EA?&'D)Gf-SAiD)3S-/$'"k$21YD?"a(+08lm"7-63d(Y&)$'GfMn>@(C.;57(+>@(1J$'*m"a$'"W57$'>@$o`O(+"a($'MiCC+b.C+32(+>\C+NCZ +T .Nj@ =kNj> j@r&J()vNCP Q!&op & p Y 9q N &'O; ()\r 8 /8()& & s;&.#1()# "R=#GF'() hZS(,(, X")F; & + F")+t ")()#1(,vNWh((ihGN6Eh%Whrq%
0L(1i P 0M_() vNts57(uU75ObC+-/;oD)GZ;+GV$.C+MO*v(?GPD)0ZC?-6"w3K57(N;oD)365WD)*v(N;$BAO(+N ?;")# &,YO5\ ();&(,\E# 7P;()\ yxOzj p )(sq N^hrh)Wh$!
0L1i 0L ;&Nw]y$BAO|+Gf-VCBD)3S-/$'""7MO>@|+*,-SgBM^(~>NMnGf3S-SA)-6>@(+"73S-/$'"7"a(+G6GV(A|;$'MOGV(+=>@(+"736CtA)-HU75WD'C+-SgBM^(CGf-SgBMO-SAO(+=S`iD.<(+"4*v|/`)-6>@(Cu3S*D)"WC+-632$'-6*v(#(+3WU7(+*,>4D)"ED)"73E>@|+=3657$BAO(C(+3TD+UiUEGf-/;oD)3Q-/$'"WC+N 86 P 8'();NX",+&(,(6:A HZ.N)
"+ P ,1i + &'N CPJ'98 N # & R F(WN AO(+"WC+-63Sb!U7(+*+3SMO*BeD)3S-/$'"N>@(+3657$BA#32$C+3SMWA)b<3K57(Y(+-P`O(+"WC+3S*,M^;+3SMn*v(Y$213Q0T$'=VU75WD'C.(^XY$'0(ogBMjD)3S-/$'"C+bC+3d(+>\C+N' + 8G xi p )sq+NOEhON%.#
"+ + iq|iEZ + Gg"WNQ Y" F(, N&7!O ")(3N&0 &'8.# N# >#1", Nj "7MO>@(+*,-/;oD)Gn>@(+3657$BA?MiC+-6"O`MUE0!-6"EA<C.;57(+>@(Ca1J$'*u3657(t*v(C.$'GfMO3Q-/$'"$21u3S0T$'=VU75WD'C.(XY$'0ZC+N^ + 8Oxn^ p )iq q+N^hON^hrq)h)Wh
"+ 8#1i Z + 8.#1",vN?7D)GV(+MO*vC#UE*v$UE*v(C4AO(CN>4D)3S*,-/;(C+N H6N ==W?HN =5=W?HN)
"+ &1i + &_;N3SMjAO(AO(C@*v|/`)-6>@(C83S*oD)"WC+-63d$'-6*v(CA|;$'MOGV(+>@(+"736C@A)-HU75WD'C+-SgBM^(C1D)-SeBGV(N*oD+UiU7$UE*,3AO(8AO(+"WC+-63d|+N 86 P f8'();N(6B&'(6:E .# &(,<&s& 2A&# N)
"+ &(ih|i + &';Nu"3K57(;$'"WC+3S*,M^;+3S-/$'"R$21NM.UE0!-6"EAkC.;57(+>@(Cu1J$'*L"a$'"7=o(ogBMn-6Gf-SeB*,-6MO>3S*D)"WC+-/(+"73C+0T$'=VU75WD'C.(!XI$'0ZC+N + F;&Y[ ZS")() Yx p h((rhq+N )!)rh
XZ (.1i = BJ'l# QDQ TZ7'") Nku"D*v$'Mr`'5~ j] C.;57(+>@(1J$'*D$'"a(+=A)-6>@(+"WC+-/$'"ED)GY3S0T$'=VU75WD'C.( >@$BAO(+G N + t[ ZR")(, ^z p h((.q+NM1)(Niq))!.(
4 ]
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Q- + (i Ntu"3657(L"7MO>@(+*+-/;oD)GC.$'GfMO3Q-/$'"3d$3S0T$<XMn-SA >@$BAO(+GHC\&.-SD Dw;(+G6G;(+"7=3d(+*v(oANlm"7-63d(8&)$'GfMn>@(8>@(+3657$BA N X& !Z7;986 0 ZS",() z p h((jq+Nf %N &%rq
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>r@ 1i @ />r# ' # P , /@r()N#b,U7(+*e+$'Gf-/;F3Q0T$'=VUE*v(CC+MO*(F>@$BAO(+GHC1J$'*3S0T$'=VU75WD'C.(!XY$'0YN^ + 8$G p )q+NEhON !(
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)((
kk
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&:;;Y#1")&' K&:;"69#1K:&'()g;Rk&f");f&J")]f v2'Y");&f;=# ()s_5 X# sJ() ()'()")M8\]
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& # (,&"324# .# "*G<#|8:;<#1()g HDB 0# '98:F#F HB",!1.() p '98:F#=>r?@Aq .#1")!0#1rT",() (,F;(,;"I'0# ")# ()&'A8\];!(6B1# s)
ut(x, t) + div(F (x, t, u(x, t))) = 0, ∀ x ∈ R
N , ∀ t ∈ R+
u(x, 0) = u0(x) ∀ x ∈ RN ,
p ! Cq
# (,V&: :;&;z# Otl")(,BG&'V I!O ",G(; 5 />r#CBG()# &' > 1NY> %ji3Nt=O")",& =OF.|iK;A# ",(,BG& OF# &' vN H# ",") ;;A@;&J(, @ ((|iWNI# ;");8Ok 8]''(,B1# s&",!<# (,")")# \
αhN ≤ m(p), m(∂p) ≤ 1
αhN−1 ∀p ∈ T ,
$#m(p)
;'L"$#!F;&' ;JI;\r .# R
N "$#!<#1()")",p ∈ T N m(∂p)
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p
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;α&&:'s")#D&:;\")# &(6:r U<#1()")")# \ / /8Gk 8];;r'&")#T (,()F()(6($#1")
u0(x)+")# +() Ot @()" Ci3N
(i) u0(x) ∈ L∞(RN) :()"v t(,' pK N'0/q ∈ R
2 ")g A ≤ u0 ≤ B6w
(ii) F ∈ C1(RN × R+ × R)
+ ∂F∂s
;'r")G0#1")s f(,98(,;();
(iii) divxF (x, t, s) =N∑
i=1
∂Fi
∂xi(x, t, s) = 0
I &O(x, t, s) ∈ (RN × R
+ × R)
(iv)k&;F# ;
K ∈ R,(,"v+tO()_
VK <∞ ;"g |∂F∂s| ≤ VK
k&&;gH
(x, t, s) ∈ (RN × R+ ×K). p ! h q
&!\;& A&'J")];F1NOk&M :'1.()&")#D ")()F v24 # U()g1N(,"I;'/:;;_5# (,&A 24(,&G (,&A")#T",(,V;&()g!g(_"$#DBG&# ()!'")()<8O'()g.# &'F(I");",(, # ()J", f#<&'()];&T :'() v24 tO()';E;! v2w(,(6:T %"$#<'")O() s&'(,gG &J",] p ! *jqu'< nm= [? LQ!&'8j CB Q!&7q(|i( ;<8GI 8]'# ()'sF")8GI 8]' p ! h qE#0B h")#x; (6(,
F ∈ C3(RN × R+ × R,RN)
N"$# :'() 24# (,U&T"$#h&:;\")# &(,#1()z.# &# JI ")()gF p ! CqP# u");u# &5(,"); H@ !1JvN @ ! #1iY>) GF# & vN *# ")",++> @;&J(, s!F &:D"32w+tO()5;E;A"32w();(,:T u")#<",(,;s& ()gu .#1!")%#
F (x, t, s) = v(x, t)f(s)#0B
v ∈ L∞ ∩ C1(RN × R+,RN)
;f ∈ C1(R,R)
N ()",sJ;h%&':'",#1;h.# '# sn "$#h")(,F(6F&E 98:F#xB ") 1.(, + ;F&':'",#1u'& F\:;:&# ")(,:uk&u +()
F (x, t, s)g(B :&( 1.H")8GI18]; p ! Hhq.# &r@(,")")# ()&'; @(," N @(," Ci(X",")H#
# '(kFs&: L_()F#1() 0 5 # ()J",/F# ;LF[ .# [")0#1L\:;:&# "3Ns+%;'()F#1(, 0 r5 &_Dh;FA .# ")T# A$#
F (x, t, s) = F (x, s) ;A98:;<#1
v2w& &'E Ot x v24 & &'")!:;");B :EsD:+:E:; (,:!.# &H= f?G")", ?G %jiY;H.# &P Q!& ;&NR= K?G",")u+NK>r9#- .# Q?>!CiLk&",&J",] p ! *jqA .# H")E# F (x, t, s) = F (s)
vPA# Q?> !Nf?G !jiY");D#1;&H2w()s:;&;;sunU"$#U B &'\; u'98:F# E <B ")F; 1.(,% v2w& &'U:",;B:Uk&T uF# ()","$# \ m_&;&:;NY&: 5\",()&'Ax()&'&:;\")(,&' IPA# ?G %1iX= ?G",")% u ;'()F#1(,A 24;&&;&s&'",(,h#1&G98:;T;",(,;s& ()g &A;"$#()"f()")(,");;'(,<#|() %? ? OQ!+'jB Q.q%1ik+()" &'\s# &' r")r98:;<#T;F&:;\"$#1&()#1(,E.# &# JI ")()g *&J")]!8GI;&Jk")(,gG!PA# T98# (,&D!#1")") # ()&'H"W2 #1.# "G'TF#18:;<#|()g 98:;<#V#1OtB ")F;1.(,k&")r&'J")];F;8 F\ ];A# k",:%=O>?@ .#1",D#
2DkPA# T#
"$# ;()7 .Gt8$G'()gF = (f1, f2)
N ;")T'98:F#U=>r?@ 2w:;&(, T")#FF# (,]&(6B1# s
W nij =
1
2
(W n
i +W nj
)−
αnij
2Snij
(F (W n
j ) − F (W ni ))· ηij
W n+1i = W n
i − ∆tn
Ai
∑j∈Ni
|eij|g(W n
i ,Wnj , S
nij, ηij
)
W 0i = 1
Ai
∫Ciu(x, 0)dx
p ! ".q
#0BCi
")AB ")! & ", N
)(ih
N v ` f B
|∂Ci|.",!I:;&()];&A *B ")! & ", N
Ai
."324# (,& *B ")! ")Ci
Neij
."W2 # &'];!%! Ci
+Cj
NV (Ci) = Cj ∈ T ;")g
Ci ∩ Cj = eijN
Ni = j/Cj ∈ V (Ci)N
aj∈Ni = (aj)j∈Ni
NSn
ij = |ηx|max(|f ′
1(Wni )| ,
∣∣f ′1(W
nj )∣∣)+ |ηy|max
(|f ′
2(Wni )| ,
∣∣f ′2(W
nj )∣∣) ,
g;'r"$# +(,U .Ot-:&'()g :H.# &*
g(W n
i ,Wnj , S
nij, ηij
)= F
(W n
ij
)· ηij
$#ηij =
(ηx
ηy
) _Y"$#r&'<# ",X&_9# s1Nαn
ij
_")[.# &# ];&'M Ls& ",!1PA# R &'# .# "*G[D;'() ];&[ L")f");R8Gk 8];; p !/hqv'&R")# ;() .OtH+S",f8GI 8]': :;( 5 ;&M")F# (,")"$#1\!Y'(,A/# ")",[98&'98&",M; (6(,M'&",.# &# F]+&' ;s& ")
αnij
;H",# ∆t
p ; (6(,x E'9#1J()",(,:0qk&g"$# ;(,U .Ot-:&'()g!&'k+H;&'9#1();/&&():+:;r;FA")#%F ()1NO",&(,(,I! 7F#|tO()%7NG"$#%;;&'B1#1(,BG(,:!;r"$#u (,'9#1
(i) g(u, v, Sn
ij, ηij
) _;&(,# s# &/&# I&_rnu;r :;&'()'# sr.# &I &'rn
vk/ (u, v) ∈ [A,B]2.
(ii) g(u, v, Sn
ij, ηij
)= −g
(v, u, Sn
ji, ηji
) IOr(u, v) ∈ [A,B]2.
(iii) g(u, v, Sn
ij, ηij
) 'r",()'98(,();U&[A,B]2
#0B;D")#%F^;F!;_9# s
")(,98(,u;v. p ! 'sq
=OTT (,(,"32w_(,<#1()x v2w&'&;&%\:;:&# ");F;D&'B:; p ;o&'FL1 qH_
A"W24&' & h
1
4
;M"$#T'")()<# &'O98:A :!# &M;!98:;<#%;sB&'\ # (,J");F;B;&!"32w(,g% ")()7s&'()gT .#
Lp k& 1 ≤ p < ∞ PA#1!+T.# &'()H :+&'F(,")#V (,(,x '9#1J()",(,:DI &")E98:;<#- F>r# CBx+D;",(Y 98:;<#7=>r?@ +"$#V ()D x98:;<#7=>r?@ .#1 0#1Nvk&! ; G']*")(,:0# (,&;8Ok&'JI ")()g;* ()'()U Ot- 24;.# ;!
)(.
B v v ]
% c-d . #/ e !$ d d " , "$ c 0 .-dHc / d #/ d);d 0 +
! "#%$&('
A< (, ]&'")r&'J")];F8GI;&Jk")(,g<",():# ()&'/0#1"$# (,& .# L"32w'.# r ()O5(,- Ot7#0B H (,(,U()(,()# ")r(,B1# s
∂u
∂t+ div(F (u)) = 0, ∀x ∈ R
2 , ∀t ∈ R+,
u(x, 0) = u0(x), ∀x ∈ R2,
p ! ! q
#0BF (u) = (f1(u), f2(u))
$#f1
;f2
% Ot +() T RB;&
R V;"$# '
C1 NO sr"); :&'(,B :r\s# &' s ;()\ /;'# s0NF#'&':;/; u : R2 × R+ → R
98:;<## OtVB",F; 1.()/k&/;D&J",]AH :; (, H")#%<#1()];&!(,B1# s
∂u
∂t+ div(F (u)) = 0 ⇐⇒
∫ tn+1
tn
∫
Ci
∂u
∂t+ div(F (u))dxdt = 0,
x(,")(, NI# ",&"324# &'CtO(,<#1()7 u .# !",% F# ()
[tn, tn+1[×Ci
;A")T8:&'] &';77# &#
Ai
un+1i − un
i
∆tn+
∫
∂Ci
F (u) · ηijdσ = 0
⇐⇒ Ai
un+1i − un
i
∆tn+∑
j∈Ni
∫
eij
F (u) · ηijdσ = 0
⇐⇒ un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij| g(un
i , unj , ηij
) p ! "%q
#0Bηij
_%")#* &F# ")E &'9#1<+g(un
i , unj , ηij)
_%"$# +(,y .Oty:&(,g :H.# &
g(un
i , unj , ηij
)' 1
|eij|
∫
eij
F (u) · ηijdσ.
=O", "32w+tO&''() z u"$# ;(,h .Gt :&(,g%x % ;! ( I:&'s!'98:F# # OtTB",F; 1.(, iAE ")M98:;<#! r>O1N !ZSZ + N !Z>G N ",98:F#H ># CBkN) ZLN
. . .
f# ;()- .Ot-:&'()g_r (,!'&_B #|(,B D'(
g(un
i , unj , ηij
)= −g
(un
j , uni , ηji
).
+ ;'T&'")#1(,-+tO&(,FgD") .Ot* Ci
B&'Cj
;'A",F^;FgH") .Ot7 Cj
B;&Ci
N#1z'()\u&]; + ;'E&'&(,:;: .#1F;# ")%# '&Eg N'(Y")%'98:F#-sB;&\1N()"X B &'\VB;&%V",(, # ()J",E o G_] p !/ !q! F"$#7",(X <;'&'B1#1() gG(
)(
N v ` f B
B:;&( 1H ");r&'")#1(, H>r# j()+5@r\() f# ;()- .Ot-:&'()g_r (,!'()_9# sH(
g (uni , u
ni , ηij) = F (un
i ) · ηij.+ ;'& &(,:;: (,") # (6g(")B1# &'($# J")Ls /B1# ",&'() ;s()g; !.# &_;r v24# & "32w()s;& # ;
eij
N.") .Gt-F:;&(,gGH (,^+&:;\s# "# .OtU8O'()g PA# ;H98# (,&!-;24(,:&''Tn%"W2 #1.# "G'A<#|8:;<#1()gA *'98:F#E >r# CBU+")!'98:F#<=O>?@ f# ;()- .Ot-:&'()g H># jBV' :.# &
g(un
i , unj , ηij
)=
1
2
(F (un
i ) · ηij + F (unj ) · ηij
)− 1
2Sn
ij(unj − un
i )
= F (uni ) · ηij +
1
2
(F (un
j ) · ηij − F (uni ) · ηij
)− 1
2Sn
ij
(un
j − uni
)
$#Sn
ij
_X"$#BG(,'/ /># CB%",O# ")[ .#1&'Y+Y v24# &' 98# gr# &]+ M"$#;")",")Ci
:;.# &
Snij = |ηx|max
|f ′
1(uni )| ,
∣∣f ′1(u
nj )∣∣+ |ηy|max
|f ′
2(uni )| ,
∣∣f ′2(u
nj )∣∣
;∆tn = tn+1 − tn
'r",.# / A/+Ai
_"324# ()&' VB ")FA !s& ")Ci
Y(,")(,# s/*k:& H N.")!'98:F# p ! '%sqM2w:;&(,r'"$# &'F!'(,B1# s"
un+1i = H
(un
i , unj∈Ni
),
p ! q q$#
unj∈Ni
s!",B1# ",&')! u # A",;",")",B(,()A
Ci
Nk;!"W24k:+5& H &")#E;",")",
i'r :.# &
H(u, vj∈Ni
)= u− ∆tn
Ai
∑
j∈Ni
|eij|g (u, vj, ηij) .
2 2 5 5, + 5 0
f# ;()- .Ot- D># CB-_;'()'# s;r;;&'B1#1(,B.#1&r;_&;(,
z Qx"!QGf>$##&%('(\C.;57|+>4D*),+-/.104C.(+*DkA)-63t>@$'"a$'3d$'"a( C+-mGo$U7|+*oD)3d(+MO* H (C+3;+*v$'-VCCBD)"73UWD)* *D+UiU7$'*,3u;5WDO;+Mn"_AO(C.(C?D)*/`)MO>@(+"736C
uni
(+3unj∈Ni
-
2 3,45 076/89:
;,/. ., z< z$'-63u0 ∈ L∞(D) ∩ L1(D)
-^MUiU7$.C.$'"WC?gBM^(?Go$'"8U7(+MO3t|;+*+-6*v( GV(C.;57|+>4D ),+-/.10C.$'M CGPD1J$'*+>@(
un+1i = un
i −∑
j∈Ni
αij(uni − un
j )
)(!
B v v ]
(+3mC+-!GV(C\;$.( ;+-/(+"736Cαij
&)|+*,- lu(+"73GV(C;$'"EA)-63S-/$'"WC
(i) αij ≥ 0U7$'MO* 32$'MO3
j ∈ Ni
!
(ii)∑
j∈Ni
αij ≤ 1!
D)GV$'*vC!U7$'Mn* 3d$'M Cn ∈ N
!t$'" D8GV(tUE*,-6"a;+-HU7( A)M>4D r-6>NMO>
minCi∈T
uni ≤ min
Ci∈Tun+1
i ≤ maxCi∈T
un+1i ≤ max
Ci∈Tun
i .
<#
un+1i = un
i −∑
j∈Ni
αij(uni − un
j )
=
(1 −
∑
j∈Ni
αij
)un
i +∑
j∈Ni
αijunj .
PT24# &] ", /8G8] (i)
(ii)
*") p ! hHhq N7#E",!&((,I 7F#|tO()
minCi∈T
uni ≤ min
Ci∈Tun+1
i ≤ maxCi∈T
un+1i ≤ max
Ci∈Tun
i
a⊥b = min (a, b)
a>b = max (a, b)
:/ < z $'-63u0
MO"a( 1J$'"a;+3Q-/$'" e+$'*,"a|( \C+MUiU7$'*,3T;$'>#UWDO;+3m(+3F = (f1, f2) :
R → R2 MO"a(m1J$'"a;+3Q-/$'"wAO(N;+GPD'CC.( C1 3d(+GgBM7(
supu∈X
|f ′1(u)| , |f ′
2(u)| = M ∈ R,
$X
(C+3uAO|Klm"7-;$'>N>@(C+Mn-63u
X = u ∈ R+/|u| ≤ ‖u0‖L∞(R2).
u"C+MUiU7$.C.(AO(?UEGfM CgBM7(GV( UWD'CAO(3d(+>#U7C∆tn
!#GV( U7|+*+-6>@c+3S*v( |∂Ci|(+3#Go D)-6*v(
Ai
AO(@GPD;(+G6GfMnGV(
Ci
&)|+*,- lu(+"73GPD4*(+GPD)3S-/$'"
∆tn ≤ Ai
|∂Ci|Sni
∀i , n ∈ N
$Sn
i = maxj∈Ni
Snij
(C+3 GPD?&.-632(CC.(AO( tM CBD)"a$'& ! ∀(i, n) ∈ N2 !ZD)GV$'*CtGV(tC.;57|+>4D\AO( YMiCBD)"a$'&
&)|+*,- lu( GV(tUE*,-6"a;+-HU7( A)M >4D r-6>NMO> GV$.;oD)G
minCi∈T
uni ≤ min
Ci∈Tun+1
i ≤ maxCi∈T
un+1i ≤ max
Ci∈Tun
i ∀Ci ∈ T , ∀n ∈ N
)(%
` B
(+3GPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-63d|L∞
||un||L∞(R2) ≤ ||u0||L∞(R2).
<#
∑
j∈Ni
|eij|g (uni , u
ni , ηij) =
∑
j∈Ni
|eij|F (uni ) · ηij
= F (uni ) ·
∑
j∈Ni
|eij|ηij
= 0,
v2 7" 98:;<#E !B " 1 ( / H> # CB ;24:&'(r "$#%F# (,]& (,B1#
un+1i = un
i − ∆tn
2Ai
∑
j∈Ni
|eij|(F (un
j ) · ηij − F (uni ) · ηij − Sn
ij
(un
j − uni
)).
A&F
L "$# C1 & [uni ⊥un
j , uni >un
j
] Ns v2 <(," +tO( νn
ij
εnij ∈
]un
i ⊥unj , u
ni >un
j
[ ;" g
F (unj ) · ηij − F (un
i ) · ηij = ~V nij · ηij
(un
j − uni
),
#0B ~V nij =
(f ′
1(νnij)
f ′2(ε
nij)
). " 2 ( Ag T" 98:<#< T> #CB ;24:&'( T")#F# (,]&
(6B1#
un+1i = un
i − ∆tn
2Ai
∑
j∈Ni
|eij|(~V n
ij · ηij − Snij
) (un
j − uni
)
=
(1 − ∆tn
2Ai
∑
j∈Ni
|eij|(Sn
ij − ~V nij · ηij
))un
i
+∆tn
2Ai
∑
j∈Ni
|eij|(Sn
ij − ~V nij · ηij
)un
j .
7
τnij =
∆tn
2Ai
|eij|(Sn
ij − ~V nij · ηij
) τni =
∑
j∈Ni
τnij.
z& F# &g Tg τnij ≥ 0
NI v2 &Ag E" 98:F# ( L∞ 9# J" NI()" v24#0B(,&
τni ≤ 1
)(iq
B ]
#
τni ≤
∑
j∈Ni
|eij|τnij
≤ supj∈Ni
|τnij|∑
j∈Ni
|eij|
≤ ∆tn
Ai
Sni |∂Ci|.
= "$# (( ∆tn ≤ Ai
Sni |∂Ci|
∀(i, n) ∈ N2 N h# &#E" H&: "#r h&'( ;()
<#jt(
& ! $ " "
! "#$! % & ')( & * +,.- / 0$#
∂u
∂t+ div (v(x, t)f (u(x, t))) = 0, ∀x ∈ R
2, ∀t ∈ R+,
u(x, 0) = u0(x), ∀x ∈ R2,
1 !"2
- div =
i=2∑
i=1
∂i 3∂i
4$5 6- &! " 4 *$ieme
7 * v(x, t)f(u(x, t))
# :9;$ieme
< xi
x ∈ R
2 v >=0
R2 × R+
- R
2 3 "$
C1 "'
div(v(x, t)) = 0
sup(x,t)∈R2×R+
|v1(x, t)|, |v2(x, t)| = V ∈ R,
f ?= ; $ C1 3
@ R
- R 3
u0
AB ? $! 9 7&? 1 !'C2 D<E F-# 1 HGJIKL ## M / N J+ O( & P Q # 1 !"2 "(R": $ < " 6-!
unij =
1
2
(un
i + unj
)−
αnij
2Snij
(f(unj ) − f(un
i ))vnij · ηij
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(un
i , unj , S
nij, ηij
)
u0i =
1
Ai
∫
Ci
W (x, y, 0)dxdy,
1 !"2
)(
` B
-
vnij =
1
|eij|
∫
eij
v(x, tn)dσ = vnji 3
Snij = max
(|f ′(un
i )vnij · ηij|, |f ′(un
j )vnij · ηij|
)3
g :$ =0 J+ 7" D <G/I:KL / &#
g(un
i , unj , S
nij, ηij
)= f
(un
ij
)vn
ij · ηij
ηij
:$ <# D αn
ij
 # D D 9 +// " +J ?- 1
αnij = αn
ji
:73 9 < 8 3 <
(i) 7
+
unij =
1
2
(un
j + uni
)−
αnji
2Snji
(f(un
i ) − f(unj ))vn
ji · ηji = unji
ηij = −ηji
g(un
i , unj , S
nij, ηij
)= f
(un
ij
)vn
ij · ηij 3#
g(un
i , unj , S
nij, ηij
)= −g
(un
j , uni , S
nij, ηji
)
O( , = + < -!6-
(ii) / 3 /5
GJ un
i = unj 3
g(un
i , uni , S
nij, ηij
)= f (un
i ) vnij · ηij 3
( <= + < D
2 3 4576/89:
:/ < z $'-63vMO"a(m1J$'"a;+3S-/$'"_AO(
R2 × R+
&)(+*vCR
2 !TAO(;+GPD'CC.( C1 !m3d(+GgBM^(
div(v(x, t)) = 0 et sup(x,t)∈R2×R+
|v1(x, t)|, |v2(x, t)| = V ∈ R,
f(C+3!MO"a(1J$'"a;+3S-/$'" AO(
R&)(+*vC
R!mAO(\;+GPD'CC.(
C1 >@$'"a$'32$'"a(!Z&)|+*,- lD)"73 |f ′(u)| ≤ M!D)&)(;
M ∈ R+
-u0
(C+3MO"a(1J$'"a;+3Q-/$'"_e+$'*,"a|(\4C+M.UiU7$'*,3I;$'>#UWDO;+3TAO$'"7"a|(!u$'" C+MUiU7$.C.(\AO(tUEGfMiCgBM^(?GV(tUWD)*D)>@c+3S*v( AO(;$'"73S*'GV(
αnij
&)|+*,- lu( GPD@*v(+GPD)3Q-/$'"
1 ≤αn
ij
Snij
∣∣f ′(anij)v
nij · ηij
∣∣ .
)(
B ]
'( UWD'CuAO(T32(+>#U7C∆tn
!aGV(U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(
Ai
A)M&)$'GfMO>@(uAO(;$'"73S* 'GV(Ci
&)|+*+- lu(+"73GPD@*v(+GPD)3Q-/$'"
∆tn ≤ 2Ai
|∂Ci|1
(1 + γni )φn
i
∀(i, n) ∈ N2,
D)&)(;
supj∈Ni
∣∣f ′(θnij)v
nij · ηij
∣∣ = φni ∀ (i, n) ∈ N
2 (+3γn
i = supj∈Ni
γnij ≥ 1
&)|+*,- lD)"73GPD@*v(+GPD)3S-/$'" C+MO-6&'D)"732(
αnij = γn
ij
Snij∣∣f ′
(an
ij
)vn
ij · ηij
∣∣ ,
GV(C.;57|+>4DL &)|+*,- lu( GV(tUE*,-6"a;+-HU7(?A)Mk>4D r-6>NMn> GV$.;oD)G
minCi∈T
uni ≤ min
Ci∈Tun+1
i ≤ maxCi∈T
un+1i ≤ max
Ci∈Tun
i
U7$'Mn* 3d$'Mn3Ci ∈ T ! n ∈ N
(+3GPD;$'"EA)-63Q-/$'"_AO(C+3/DieB-6Gf-63d|L∞
||un||L∞(R2) ≤ ||u0||L∞(R2).
< D<<G/IKL "(R" ?$ =
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(un
i , unj , S
nij, ηij
),
-
g(un
i , unj , S
nij, ηij
)= f
(un
ij
)vn
ij · ηij
unij =
1
2
(un
i + unj
)−
αnij
2Snij
(f(un
j ) − f(uni ))vn
ij · ηij.
Bf < "$
C1 3 ( /7 , 0 " 1
( +J an
ij
& :)( " -!# ]un
i ⊥unj , u
ni >un
j
[ "' (f(un
j ) − f(uni ))vn
ij · ηij = f ′ (anij
)vn
ij · ηij
(un
j − uni
),
O( ,)( D# >" ? -
unij =
1
2
(un
i + unj
)−
αnij
2Snij
f ′ (anij
)vn
ij · ηij
(un
j − uni
)
= uni +
1
2
(1 −
αnij
Snij
f ′ (anij
)vn
ij · ηij
)(un
j − uni
)
= unj − 1
2
(1 +
αnij
Snij
f ′ (anij
)vn
ij · ηij
)(un
j − uni
).
)(
` B
,
δnij =
αnij
Snij
f ′ (anij
)vn
ij.ηij,
& " ? (RD! >" ? -J
unij = un
i +1
2
(1 − δn
ij
) (un
j − uni
).
> - " # , > f(un
ij) 0- & B
uni 3
+J
θnij ∈
]un
i ⊥unij, u
ni >un
ij
[,
'
g(un
i , unj , S
nij, ηij
)= f
(un
ij
)vn
ij · ηij
=
(f(un
i ) +1
2
(1 − δn
ij
)f ′ (θn
ij
) (un
j − uni
))vn
ij · ηij.
B ,
∑
j∈Ni
|eij|f (uni ) vn
ij · ηij = f (uni )∑
j∈Ni
|eij|vnij · ηij
= f (uni )∑
j∈Ni
∫
eij
v(x, tn) · ηijdσ
= f (uni )
∫
Ci
div(v(x, tn))dx
= 0,
# / div(v(x, tn)) = 0 3
( D<<G/IKL "(R" ?$ =
un+1i = un
i − ∆tn
2Ai
∑
j∈Ni
|eij|(1 − δn
ij
)f ′ (θn
ij
)vn
ij · ηij
(un
j − uni
)
=
(1 − ∆tn
2Ai
∑
j∈Ni
|eij|(δnij − 1
)f ′ (θn
ij
)vn
ij · ηij
)un
i
+∆tn
2Ai
∑
j∈Ni
|eij|(δnij − 1
)f ′ (θn
ij
)vn
ij · ηijunj .
,
τnij =
∆tn
2Ai
|eij|(δnij − 1
)f ′ (θn
ij
)vn
ij · ηij,
B ]
F ; f
3 (
f ′ (anij
)vn
ij · ηij
f ′ (θn
ij
)vn
ij · ηij
3 O(
τnij =
∆tn
2Ai
|eij|(|δn
ij| − sgn(f ′(u)vn
ij · ηij
)) ∣∣f ′ (θnij
)vn
ij · ηij
∣∣ ,,
un+1i =
(1 −
∑
j∈Ni
τnij
)un
i +∑
j∈Ni
τniju
nj .
( # > 1 ! hHh23 D< GJIKL -#" 1 >/ " B < +J
τnij ≥ 0 et
∑
j∈Ni
τnij ≤ 1.
τnij ≥ 0 ⇐⇒
∣∣δnij
∣∣− sgn(f ′(u)vn
ij · ηij
)≥ 0
-# ∣∣δnij
∣∣ ≥ 1,
∣∣δnij
∣∣ ≥ 1 ⇐⇒ ∀(i, n) ∈ N2, ∀j ∈ Ni, ∃ γn
ij ≥ 1 '
αnij = γn
ij
Snij∣∣f ′
(an
ij
)vn
ij · ηij
∣∣1 ! )(2
∑
j∈Ni
|τnij| ≤ ∆tn|∂Ci|
2Ai
(1 + γni )φn
i ,
- supj∈Ni
|f ′(θnij)v
nij · ηij| = φn
i , ∀ (i, n) ∈ N2
γni = sup
j∈Ni
γnij.
# D un
ij
@! & (Rθn
ij
1 -# 1 !Hh/ 2 < & ?)( D# > D2
3 ( :
∆tn ≤ 2Ai
|∂Ci|1
(1 + γni )φn
i
3
∑
j∈Ni
|τnij| ≤ 1 3
( > ? " B < +J
3
? ( " H :$ Ci 3
D<< -# 1 ( " /-!
un+1i = H
(un
i , unj∈Ni
).
B : " ? O( <#$ /$# card(Ni) = 3 3
-
H(uni , uj∈Ni) = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(uni , uj, S
nij, ηij).
h
` B
( # ?$ 1 1 !Hh.23 = J+ 7" (
H " # :!# :9 D& ?-!## un
i 3 unj 3
- j ∈ Ni
B: ( #/ ? " 1 ! )(2
3)( D# > "(R"
unij =
1
2
(un
j + uni
)−γn
ij
2sgn
(f ′ (an
ij
)vn
ij · ηij
) (un
j − uni
).
/ 3 # 1/+ " 7&# <
γnij
9γ 3
& " D <G/I:KL "(R /-!
unij =
1
2
(un
j + uni
)− γ
2sgn
(f ′(an
ij)vnij · ηij
) (un
j − uni
)
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
g(un
i , unj , S
nij, ηij
).
:/ < z $'-63fMn"a( 1J$"a;+3Q-/$'" AO(w;+GPD'CC.(
C1 (+3?>@$'"a$'32$'"a(! D)GV$'*vCC.$'MiCkGPD;$'"EA)-63Q-/$'"kC+Mn-6&'D)"73d(
∆tn ≤ 2Ai
|∂Ci|ψni (γ + 1)
, ∀(i, n) ∈ N2,
D)&)(;ψn
i = supj∈Ni
∣∣f ′ (unij
)vn
ij · ηij
∣∣ ,GV(C.;57|+>4DL (C+3>@$'"a$'32$'"a( -
<+
f
3#
f ′ (anij
)vn
ij ·ηij
f ′ (un
ij
)vn
ij ·ηij
5 3
sgn(f ′(u)vn
ij · ηij
)= sij.
B " ?$ & / ( "? $ < " 6-#
unij =
1
2
((1 + γsij) u
ni + (1 − γsij) u
nj
)
∂H∂uj
(u, uj∈Ni
)= −∆tn
2Ai
|eij| (1 − γsij) sij
∣∣f ′ (unij
)vn
ij · ηij
∣∣
/: ∂H∂uj
(u, uj∈Ni
)≥ 0 3
: O( -sij − γ ≤ 0
+
γ ≥ 1,# ' 8-" 1&
∂H∂un
i
(un
i , uj∈Ni)
= 1 − ∆tn
2Ai
∑
j∈Ni
|eij| (1 + γsij)∣∣f ′(un
ij)vnij · ηij
∣∣ sij
∂H∂un
i
(un
i , uj∈Ni)≥ 0 ⇐⇒ ∆tn
2Ai
∑
j∈Ni
|eij| (sij + γ)∣∣f ′ (un
ij
)vn
ij · ηij
∣∣ ≤ 1
).
B ]
B ,∑
j∈Ni
(sij + γ)∣∣f ′ (un
ij
)vn
ij · ηij
∣∣ ≤ (γ + 1) supj∈Ni
∣∣f ′ (unij
)vn
ij · ηij
∣∣ |∂Ci|.
, supj∈Ni
|f ′(unij)v
nij · ηij| = ψn
i ,
O( ∑
j∈Ni
|eij| (γ + sij) f′(un
ij)vnij · ηij ≤ (γ + 1)ψn
i |∂Ci|.
P < P O( M , ,)( D# un
ij
1 - 1 ! h/ 2
)( D# /" / 23 O(
∆tn ≤ 2Ai
|∂Ci|1
(γ + 1)ψni
∀(i, n) ∈ N2,
# # ∂H∂un
i
(un
i , uj∈Ni)≥ 0.
$O "#J & $ "#
" (R" & > # ## & 4)( & +,.- / 0$#
∂u
∂t+ div (F (u(x, t)) = 0, ∀x ∈ R
2, ∀t ∈ R+,
u(x, 0) = u0(x), ∀x ∈ R2
1 ! .2
- F = (f1, f2)
>=Q R
- R
2 3 $
C1 u : R
2 × R+ → R
D ,-# 1 G/I:KL < / 1 !* 2 "(R" $ =# 6-!
unij =
1
2
(un
i + unj
)−
αnij
2Snij
(F (un
j ) − F (uni ))· ηij
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(un
i , unj , S
nij, ηij
)
u0i =
1
Ai
∫
Ci
u (x, y, 0)dxdy.
1 ! h#2
- Sn
ij = |ηx|max|f ′
1(uni )| ,
∣∣f ′1(u
nj )∣∣+ |ηy|max
|f ′
2(uni )| ,
∣∣f ′2(u
nj )∣∣
g :$ =0 J+ 7" / &#
g(un
i , unj , S
nij, ηij
)= F
(un
ij
)· ηij
ηij =
(ηx
ηy
) : <# D
` B
:73 9 < 8 3 <
D<<G/IKL -!= D :&#
2 3 4576/89:
:/ < z $'-63F = (f1, f2)
MO"a(Z1J$'"a;+3S-/$'" AO(R&)(+*vC
R2 !!AO( ;+GPD'CC.( C1 !3d(+G6GV(gBM^(
supu∈R
|f ′1(u)|, |f ′
2(u)| ≤ M ∈ R+ - u0
(C+3tMn"a(1J$'"a;+3Q-/$'"e+$'*,"a|(LDC+MUiU7$'*,3;$'>#UWDO;+3AO$'"7"a|($'" C+MUiU7$.C.(?gBM^(?GV(uUWD)*D)>@c+3S*v( AO(N;$'"73Q* 'GV(
αnij
&)|+*,- lu( GPD8*v(+GPD)3Q-/$'"
1 ≤αn
ij
Snij
∣∣∣~V nij · ηij
∣∣∣ ,
GV(UWD'CNAO(\32(+>#U7C∆tn
!uGV(U7|+*,-6>@c+3S*(L(+3tGo D)-6*v(Ai
A)Mw&)$'GfMO>@(NAO(4;$'"73S*'GV(Ci
&)|+*,- lu(+"73uGPD*v(+GPD)3Q-/$'"
∆tn ≤ 2Ai
|∂Ci|1
(1 + γni )φn
i
,D)&)(;
φni = sup
j∈Ni
∣∣∣~Unij · ηij
∣∣∣ ∀ (i, n) ∈ N2 (+3
γni = sup
j∈Ni
γnij ≥ 1
D)&)(; ~Unij
(+3 ~V nij
C.$'"73TAO|Klm"7-VC AiD)"WC GPD?UE*v(+MO&)((+3γn
ij
&)|+*,- lu( GPD@*v(+GPD)3S-/$'" C+Mn-6&'D)"73d(
αnij = γn
ij
Snij∣∣∣~V n
ij · ηij
∣∣∣,
1 ! ).2
D)GV$'*vCN$'"_D@GV(uUE*,-6"a;+-HU7( A)Mk>4D r-6>NMn> GV$.;oD)G
minCi∈T
uni ≤ min
Ci∈Tun+1
i ≤ maxCi∈T
un+1i ≤ max
Ci∈Tun
i ,
U7$'Mn* 3d$'Mn3Ci ∈ T (+3U7$'Mn* 3d$'Mn3
n ∈ N(+3GPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-632|
L∞
||un||L∞(R2) ≤ ||u0||L∞(R2).
< D< >- 1 :GJIKL ( " =
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(un
i , unj , S
nij, ηij
)
-
g(un
i , unj , S
nij, ηij
)= F
(un
ij
)· ηij
unij =
1
2
(un
i + unj
)−
αnij
2Snij
(F (un
j ) − F (uni ))· ηij.
!
B ]
BF
: $ C1 3
( : ( +J an
ij
bnij ∈
]un
i ⊥unj , u
ni >un
j
[ "' (F (un
j ) − F (uni ))· ηij = ~V n
ij · ηij
(un
j − uni
),
1 ! C2
- ~V nij =
(f ′
1(anij)
f ′2(b
nij)
)3 O( ,)( D# > ? -
unij =
1
2
(un
i + unj
)−
αnij
2Snij
~V nij · ηij
(un
j − uni
)
= uni +
1
2
(1 −
αnij
Snij
~V nij · ηij
)(un
j − uni
).
, δnij =
αnij
Snij
~V nij · ηij 3
& " ? (RD! >" ? -J
unij = un
i +1
2
(1 − δn
ij
) (un
j − uni
).
& 7 - " 4 < F (un
ij) - & <
uni 6-! #
ηij 3 +J
θnij
εnij ∈
]un
i ⊥unij, u
ni >un
ij
[3 " 8
g(un
i , unj , S
nij, ηij
)= F
(un
ij
)· ηij
=
(F (un
i ) · ηij +1
2
(1 − δn
ij
)~Un
ij · ηij
(un
j − uni
)),
- ~Unij =
(f ′
1(θnij)
f ′2(ε
nij)
)
B ,∑
j∈Ni
|eij|g(un
i , uni , S
nij, ηij
)=
∑
j∈Ni
|eij|F (uni ) v
nij · ηij
= F (uni )∑
j∈Ni
|eij| · ηij
= 0, O(
un+1i = un
i − ∆tn
2Ai
∑
j∈Ni
|eij|(1 − δn
ij
)~Un
ij · ηij
(un
j − uni
)
=
(1 − ∆tn
2Ai
∑
j∈Ni
|eij|(δnij − 1
)~Un
ij · ηij
)un
i
+∆tn
2Ai
∑
j∈Ni
|eij|(δnij − 1
)~Un
ij · ηijunj .
)%
` B
,
τnij =
∆tn
2Ai
|eij|(δnij − 1
)~Un
ij
=∆tn
2Ai
|eij|(∣∣δn
ij
∣∣− sgn(~Un
ij · ηij
)) ∣∣∣~Unij · ηij
∣∣∣ ,,
un+1i =
(1 −
∑
j∈Ni
τnij
)un
i +∑
j∈Ni
τniju
nj .
( # > 1 ! hHh25 D <G/I:KLM- 1 >" < +J
τnij ≥ 0
∑
j∈Ni
τnij ≤ 1.
τnij ≥ 0 ⇐⇒
∣∣δnij
∣∣− 1 ≥ 0-! ∣∣δn
ij
∣∣ ≥ 1,
δnij ≥ 1 ⇐⇒ ∀(i, n, j) ∈ N
2 ×Ni ∃ γnij ≥ 1
' αn
ij = γnij
Snij∣∣∣~V n
ij · ηij
∣∣∣
∑
j∈Ni
|τnij| ≤ ∆tn|∂Ci|
2Ai
(1 + γni )φn
i ,
- supj∈Ni
∣∣∣~Unij · ηij
∣∣∣ = φni , ∀ (i, n) ∈ N
2 γn
i = supj∈Ni
γni .
( ∆tn ≤ 2Ai
|∂Ci|1
(1 + γni )φn
i
3, ∑
j∈Ni
|τnij| ≤ 1
( H D H/ "
+/
3
? ( " H :$ Ci 3
D<< -# 1 ( " /-!
un+1i = H
(un
i , unj∈Ni
),
-
H(un
i , uj∈Ni)
= uni − ∆tn
Ai
∑
j∈Ni
|eij|g(u, uj, S
nij, ηij
).
q
B ]
( # F /'1 1 !Hh* 23 D PG/IKL ? ( " H
" :# :9 D& 8-!#$# un
i , unj∈Ni
B: ( #/ ? <" 0 " > < +J 1 ! .C2
3 "
αnij = γn
ij
Snij
|~V nij · ηij|
- γn
ij ≥ 1.
@ (R C# 1 ! C23)( D# > ? -
unij =
1
2
(un
j + uni
)−γn
ij
2sgn
(~V n
ij · ηij
) (un
j − uni
).
/ $ #7 3 # "
γnij = γ ∀(i, n, j) ∈
N2 ×Ni
:/ < z $'-63F = (f1, f2)
MO"a(!1J$'"a;+3Q-/$'"wAO(N;+GPD'CC.(C1 !TD)&)(; f1
(+3f2
AO(+M1J$'"a;+3Q-/$'"WC>@$'"a$'32$'"a(C!mD)GV$'*CC.$'MiC GPDk;$'"EA)-63S-/$'" C+MO-6&'D)"732(
∆tn ≤ 2Ai
|∂Ci| (γ + 1)ψni
∀(i, n) ∈ N2
D)&)(;ψn
i = supj∈Ni
∣∣F ′ (unij
)· ηij
∣∣ (+3γ ≥ 1
GV(C.;57|+>4DL (C+3>@$'"a$'32$'"a( -
<+ > J+ =
f1
f2
$ 6- F
"# 6-! > /
ηij 3# ~V n
ij · ηij
F ′(un
ij) · ηij
: 5,
sgn(~V n
ij · ηij
)= sij 3
#
unij =
1
2
(un
j + uni
)− γsij
2
(un
j − uni
)
=1
2
((1 + γsij)u
ni + (1 − γsij)u
nj
)
∂H∂uj
(un
i , uj∈Ni)
= −∆tn
2Ai
∑
j∈Ni
|eij| (1 − γsij) sij
∣∣F ′ (unij
)· ηij
∣∣
/: ∂H∂uj
(un
i , uj∈Ni)≥ 0 3
O( -sij − γ ≤ 0 ⇐⇒ sij ≤ γ
+
γ ≥ 1 3 " ?-#9
∂H∂un
i
(un
i , uj∈Ni)
= 1 − ∆tn
2Ai
∑
j∈Ni
|eij| (sij + γ)∣∣F ′ (un
ij
)· ηij
∣∣
)
` B
∂H∂un
i
(un
i , uj∈Ni)≥ 0 ⇐⇒ ∆tn
2Ai
∑
j∈Ni
(sij + γ)∣∣F ′ (un
ij
)· ηij
∣∣ ≤ 1
B ,
∑
j∈Ni
(sij + γ)∣∣F ′ (un
ij
)· ηij
∣∣ ≤∑
j∈Ni
|eij| (1 + γ)∣∣F ′(un
ij) · ηij
∣∣
≤ (1 + γ) supj∈Ni
∣∣F ′ (unij
)· ηij
∣∣ |∂Ci|.
, ψn
i = supj∈Ni
∣∣F ′ (unij
)· ηij
∣∣ , ( /:
∑
j∈Ni
(1 + γ)∣∣F ′ (un
ij
)· ηij
∣∣ ≤ (1 + γ)ψni |∂Ci|.
( ∆tn ≤ 2Ai
|∂Ci|1
(1 + γ)ψni
∀n ∈ N,# ∂H
∂uni
(un
i , uj∈Ni)≥ 0.
J/ " $ $"
< &# 9 (R / = # & B / 3 ;/
# B (R "< $! un
ij
" 9 D&! ; D >< un
ij$ $ /-!
unij ∈
[un
i ⊥unj , u
ni >un
j
].
B
unij =
1
2
(1 + δn
ij
)un
i +1
2
(1 − δn
ij
)un
j ,
/ 1 + δn
ij ≥ 0
1 − δnij ≥ 0 3
"( ∣∣δnij
∣∣ ≤ 1 3 A" 6-! ;9
αnij ≤
Snij∣∣∣~V n
ij · ηij
∣∣∣ + "$, - $,/ 7 /
# ∣∣δnij
∣∣ ≥ 1 3
αnij =
Snij∣∣∣~V n
ij · ηij
∣∣∣
% # D G/I:KLBG ' " 1 !(
B ]
D QG/I:KL = # - F ′ · ηij
& B< /" / 2 "(R :$ = -!
unij =
1
2
(un
i + unj
)− 1
2∣∣∣~V n
ij · ηij
∣∣∣
(F (un
j ) − F (uni ))· ηij
un+1i = un
i − ∆tn
Ai
∑
j∈Ni
|eij|g(uni , u
nj , S
nij, ηij)
u0i =
1
Ai
∫
Ci
u(x, y, 0)dxdy,
1 2
unij =
1
2
(un
i + unj
)− 1
2sgn
(~V n
ij · ηij
) (un
j − uni
)
un+1i = un
i − ∆tn
Ai
∑j∈Ni
|eij|g(un
i , unj , S
nij, ηij
)
u0i = 1
Ai
∫Ciu(x, y, 0)dxdy.
1 2
! " O
H / H "# 1 q2 - / $# 3 /
<&# L /&# L / " D< <- 1 3 - <
3 <#$
∂u
∂t+∂u
∂x+ 2
∂u
∂y= 0, (x, y) ∈ D, t ∈ R+
u(x, y, 0) = u0(x, y) = cos2[2π((x− 0.5)2 + (y − 0.25)2
)],
1 q#2
- D = [0, 2] × [0, 1]
/A # " 1D
-! : yj 3
"< @)( 8 $ # .- " >#
3 ( # " =< &!$
x
# -3 B)( # &#$;-#! / $ & /
& ? "
Hmax B & ? #O ( : &#
Hmax = maxi
(maxj∈Ni
|eij|)
? D 7" ?= 8&#? D<<G/IKLBG : 6-!
h
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25maillage 150x60
$ % ()/ % n/) n & "0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1condition initiale,iso−concentration
X
Y
Hh .*W"() # 5 # n
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1solution numrique−iso−concentration
X
Y
/) #"H / %& $/ ∆t = 0.00047
% O !
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1solution exacte−iso−concentration
X
Y
#" $ /) %)" % !
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u(t
)
&'()+*-,*/.103246587:9<;>=?7A@CBEDF2G@4?HFI2JKJL4NMPO%QRICST467
−3.6 −3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8−7
−6.5
−6
−5.5
−5
−4.5
Log(Hmax)
Lo
g(L
1−
err
eu
r)
pente=1.17
&'()U*V,WX.0324NQRYZ7=\[ 7<QFQR7%46Q]BD^GDF2L_%IO<7`L5Z7%JaHF7
' 1.17
bNc-b
% . ! #/ !$ " , "$ ! 0 .
/ . " ) $
B Q&# ; "< D#/ </ D< I .- " /A D< G/I:KL # ; 0 J+ ( & Q ! & ?)( & > + - 0$# -!
∂W
∂t+ div(F (W )) = 0 ∀x ∈ R
2 , ∀ t ∈ R+,
W (x, 0) = W0(x) ∀x ∈ R2,
1 "! 2
- F (W ) = (F1(W ), F2(W ))
F1
F2
* + = R
m - " R
m , m ≥1 $
C1 W : R
2 × R+ → Rm D<; B-# $# @ 8
1 "! 2 ( ": $ < " -!
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
|eij|g(W n
i ,Wnj , ηij
)
g(W n
i ,Wnj , ηij
) :$ =Q J+ 7" / &#
g(W n
i ,Wnj , ηij
)=
1
|eij|
∫
eij
F (W ) · ηijdσ.
# .+J " D<7 J+0- %# / "
! "#%$&('
;= Q J+ 7" D I - &#
g(W n
i ,Wnj , ηij
)=
1
2
(F (W n
i ) · ηij + F (W nj ) · ηij
)− 1
2Sn
ij
(W n
j −W ni
),
1 2
- Sn
ij
$0- <I .- ! 7&# O( < D&# # 7 /
Ci
&#
Snij = max
(max
1≤p≤m
∣∣∣[λni ]p
∣∣∣ , max1≤p≤m
∣∣∣[λn
j
]p
∣∣∣),
[λn
i ]p [
λnj
]p
< p−
-!# </ < H ∇F (W ni ) · ηij - ∇F (W n
j ) · ηij
∆tn = tn+1 − tn
: >& ? < Ai
)( ! Q- >
Ci
Q-#"&# B= @ $ = 0 J+ 7" D " -!6-
hrh
:/ < Sx$'-63F : R
m → R2 MO"a(E1J$'"a;+3Q-/$'"NAO(I;+GPD'CC.( C1 !(+3 W0
Mn"a(^1J$'"a;+3S-/$'"AiD)"WC
L∞(R2)AO$'"7"a|( -u"LC+MUiU7$.C.(tAO(ZUEGfMiCIgBM7(YGV( UWD'CIAO(t3d(+>#U7C
∆tn!GV( U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(
Ai
AO( GPDk;(+G6GfMOGV(Ci
&)|+*,- lu(+"73mGPD@*v(+GPD)3Q-/$'"
∆tn ≤ Ai
|∂Ci|Sni
, ∀i, n ∈ N,
$Sn
i = supj∈Ni
Snij
(C+3GPD4&.-632(CC.(\AO(tM CBD)"a$'& -u" D !D)GV$'*vC GPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-632|
L∞
||W n||L∞(R2) ≤ ||W0||L∞(R2)'D@"a$'*+>@(L∞ U7$'MO* Mn"w&)(;+3d(+MO*
X = (X1, . . . , Xm) ∈ Rm (C+3TAO$'"7"a|(uUWD)*
||X||L∞ = sup1≤r≤m
|Xr|.
< ∑
j∈Ni
|eij|F (W ni ) · ηij = 0,
# ;" < D< 7- #
1 25 = -!
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
|eij|(g(W n
i ,Wnj , ηij
)− F (W n
i ) · ηij
)
= W ni − ∆tn
2Ai
∑
j∈Ni
|eij|[(F (W n
j ) − F (W ni ))· ηij − Sn
ij
(W n
j −W ni
)].
< "# B I & +/ A(W n
i ,Wnj ; ηij
)
W ni
W n
j
/-! 7$ ηij
-#"&# # < 8 /-! RI !
(i) A (W ni ,W
ni ; ηij) = A (W n
i ; ηij)
(ii) A(W n
i ,Wnj ; ηij
)(Wj −W n
i ) =(F (W n
j ) − F (W ni ))· ηij
(iii) A(W n
i ,Wnj ; ηij
) /$ &# #/ 7& 7 ;=< &# - */ [Rn
ij
]p
p = 1, . . . , m
@ J+ -!# 4 [λn
ij
]p
/
p = 1, . . . , m
D< I .- "(R :# 8 $ < " -!
W n+1i = W n
i − ∆tn
2Ai
∑
j∈Ni
|eij|[A(W n
i ,Wnj ; ηij
) (W n
j −W ni
)− Sn
ij
(W n
j −W ni
)]
=
[Im − ∆tn
2Ai
∑
j∈Ni
|eij|(−A
(W n
i ,Wnj ; ηij
)+ Sn
ijIm)]
W ni
+∆tn
2Ai
∑
j∈Ni
|eij|(−A
(W n
i ,Wnj ; ηij
)+ Sn
ijIm)W n
j ,1 h2
hZ
Im : B > Mm(R)
7 * '
W ni 3 W n
j
W n+1
i
& '@& < ([Rnij
]p
)1≤p≤m
3,
W n+1i =
p=m∑
p=1
[αn+1
i
]p
[Rn
ij
]p, W n
i =
p=m∑
p=1
[αni ]p[Rn
ij
]p,
W n
j =
p=m∑
p=1
[αn
j
]p
[Rn
ij
]p,
O( , D 1 h2 :/-!# :9
p=m∑
p=1
[[αn+1
i
]p−(
1 − ∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
))[αn
i ]p
][Rn
ij
]p
−p=m∑
p=1
[∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
) [αn
j
]p
][Rn
ij
]p
= 0.
[Rnij
]p
/ " p = 1, . . . , m 3
# #
p=m∑
p=1
[[αn+1
i
]p−(
1 − ∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
))[αn
i ]p
][Rn
ij
]p
−p=m∑
p=1
∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
) [αn
j
]p
[Rn
ij
]p
= 0
[αn+1
i
]p−(
1 − ∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
))[αn
i ]p
− ∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
) [αn
j
]p
= 0,
p = 1, . . . , m
⇐⇒[αn+1
i
]p
=
(1 − ∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
))[αn
i ]p
+∆tn
2Ai
∑
j∈Ni
|eij|(−[λn
ij
]p+ Sn
ij
) [αn
j
]p,
p = 1, . . . , m
, [
γnij
]p
=∆tn
2Ai
|eij|(−[λn
ij
]p+ Sn
ij
) p = 1, . . . , m
h8"
, <# [γn
ij
]p≥ 0
p = 1, . . . , m
# / " / 0$ D# L∞ 3
7 /- ;$ 8 ∑
j∈Ni
[γn
ij
]p≤ 1
p = 1, . . . , m.
H
∑
j∈Ni
[γn
ij
]p
=∑
j∈Ni
∆tn
2Ai
|eij|(−[λn
ij
]p+ Sn
ij
)
≤ ∆tn
Ai
Sni
∑
j∈Ni
|eij|
≤ ∆tn
Ai
Sni |∂Ci|,
Sn
i = maxj∈Ni
Snij
( ∆tn ≤ Ai
Sni |∂Ci|
∀ (i, n) ∈ N2,
, ∑
j∈Ni
[γn
ij
]p≤ 1,
p = 1, . . . , m 3
O( , / 0 D# L∞
$ " " M2D
2 4 <5 89/
" " > "# /-!
∂W
∂t+ A
∂W
∂x+B
∂W
∂y= 0
W (x, y, 0) = W0 (x, y) ,
1 h 2
- A
B
J+ 8 & Mm (R)
" / " 3 ( / L#
J (ηij) = A · (ηij)x +B · (ηij)y
$ &# ! R 3
! m
-!# # ([λij]p
)1≤p≤m
J+
- # ([Rij]p
)1≤p≤m
D<E F-# # HGJIKL ( " /, 1 h 20 < "
h
6-!
W nij =
1
2
(W n
i +W nj
)−
αnij
2Snij
(A · (ηij)x +B · (ηij)y)(W n
j −W ni
)
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
|eij|g(W n
i ,Wnj , S
nij, ηij
)
W 0i =
1
Ai
∫
Ci
W (x, y, 0)dxdy,
1 hrh#2
- g(W n
i ,Wnj , S
nij, ηij
)= J (ηij) ·
(W n
ij
)
ηij
:$ <# D Sn
ij
:$ -J I .- "# >&# ? O( > D# # > >$ " Ci
&#
Snij = max
1≤p≤m
(∣∣∣[λij]p
∣∣∣)
:/ < z$'-63W0
Mn"a(1J$'"a;+3S-/$'"yAiD)"WCL∞ (
R2 × R+
) AO$'"7"a|(t$'" C+M.UiU7$.C.(gBM^(?GV(tUWD)*D)>@c+3S*v( AO(;$'"73S*'GV(
αnij
&)|+*,- lu( GPD@*v(+GPD)3Q-/$'"
minj∈Ni
αn
ij
≥ max
j∈Ni
Sn
ij
[λij]p
.
'( UWD'CuAO(T32(+>#U7C∆tn
!aGV(U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(
Ai
A)M&)$'GfMO>@(uAO(;$'"73S* 'GV(Ci
&)|+*+- lu(+"73
∆tn ≤ 2Ai
|∂Ci|maxj∈Ni
∣∣∣[λij]p
∣∣∣(1 +
∣∣δnij
∣∣)
D)&)(;δnij =
αnij
Snij
[λij]p ,D)GV$'*vC\GV( C.;57|+>4DAO(\&)$'GfMn>@(Clm"7-VC? ),+-0D+UiUEGf-SgBM7|IU7$'Mn*
GV(C+bC+3dc+>@($),+-(04&)|+*,- lu( GPDk;$'"EA)-63S-/$'"wAO(C+3/DieB-6Gf-63d|L∞ C+Mn-6&'D)"73d(
||W n||L∞(R2) ≤ ||W0||L∞(R2).
<P 7
W ni , W n
j , W nij
W n+1
i
& & =7 7&# - # : J (ηij)
/-!
W ni =
p=m∑
p=1
[ξni ]p [Rij]p , W n
j =
p=m∑
p=1
[ξnj ]
p[Rij]p , W n
ij =
p=m∑
p=1
[ξnij]p [Rij]p
W n+1
i =
p=m∑
p=1
[ξn+1i ]p [Rij]p
h
(R# >/" / ? D/< G/I:KL 1 h6" 25 3
p = 1, . . . , m
[ξnij]p =
1
2
([ξn
i ]p + [ξnj ]
p
)−
αnij
2Snij
[λij]p
([ξn
j ]p− [ξn
i ]p
).
1 hZ2
, δnij =
αnij
Snij
λij 31 HhZC25 -J
[ξnij]p =
1
2
(1 + δn
ij
)[ξn
i ]p +1
2
(1 − δn
ij
)[ξn
j ]p.
∑
j∈Ni
|eij|J (ηij)Wni = 0 3
# # 5 0 > (RD! /8 D
G/I:KL 1 Hh8" 25 ?$ & >=7 >#? @- / / "(R
[ξn+1i ]p = [ξn
i ]p −∆tn
Ai
∑
j∈Ni
|eij| [λij]p
([ξn
ij]p − [ξni ]p
)
= [ξni ]p −
∆tn
Ai
∑
j∈Ni
|eij| [λij]p
(1
2
(1 + δn
ij
)[ξn
i ]p +1
2
(1 − δn
ij
)[ξn
j ]p− [ξn
i ]p
)
=
(1 −
∑
j∈Ni
γnij
)[ξn
i ]p +∑
j∈Ni
γnij[ξ
nj ]
p,
- γn
ij =∆tn
2Ai
|eij| [λij]p(−1 + δn
ij
) ( # 5 : 1 Hhh#2* "A$ / #
L∞ 3 5
"? γn
ij ≥ 0 ∑
j∈Ni
γnij ≤ 1
γnij ≥ 0 ⇐⇒ [λij]p
(−1 + δn
ij
)≥ 0 ⇐⇒ −sij +
∣∣δnij
∣∣ ≥ 0-# ∣∣δn
ij
∣∣ ≥ 1,
- sij = sgn
([λij]p
) H# & :# @ 7 8$ 0 ?# " ∑
j∈Ni
γnij ≤ 1
H∑
j∈Ni
γnij ≤
∆tn
2Ai
|∂Ci|maxj∈Ni
∣∣∣[λij]p
∣∣∣(sij +
∣∣δnij
∣∣),
O( ?$ 0 D!
∆t ≤ 2Ai
|∂Ci|maxj∈Ni
∣∣∣[λij]p
∣∣∣(sij +
∣∣δnij
∣∣)
D< GJIKL -#"= $ 0 D!L∞
hrq
2 4 <5 "8/9
D/<P Q- # G/IKL 7 1 ! 2 "(R" H < " 6-!
W nij =
1
2
((W n
i +W nj
)−αn
ij
Snij
(F (W n
j ) − F (W ni ))· ηij
)
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
|eij|g(W n
i ,Wnj , S
nij, ηij
)
W 0i =
1
Ai
∫
Ci
W (x, y, 0)dxdy,
1 h8"C2
g $ =Q J+ 7" D<<G/IKL / &#
g(W n
i ,Wnj , S
nij, ηij
)= F
(W n
ij
)· ηij
Sn
ij = max1≤p≤m
(∣∣∣∣[λn
ij
]p
∣∣∣∣)
ηij
7 <# # D [λn
ij
]p
7 -!# ; # 7 0$ < I RI ! / # & B D I .- ; BD
W ni
W n
j 3 6-!
$; ηij
;= Q J+ 7" D <G/I:KL D "-!-
:/ < $'-/(+"73F = (F1, F2)
MO"a(1J$'"a;+3S-/$'" AO(R
m &)(+*vCR
2 !tAO(N;+GPD'CC.( C1(+3W0
Mn"a(m1J$'"a;+3S-/$'"_AiD)"WCL∞ (
R2) AO$'"7"a|( u$'" C+M.UiU7$.C.( gBM^( GV(tUWD)*D)>@c+3S*(?AO(N;$'"73Q* 'GV(
αnij
&)|+*,- lu( GPD8*v(+GPD)3S-/$'"
minj∈Ni
αn
ij
≥ max
j∈Ni
Snij∣∣∣∣
[λn
ij
]p
∣∣∣∣
.
'( UWD'CuAO(T32(+>#U7C∆tn
!aGV(U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(
Ai
A)M&)$'GfMO>@(uAO(;$'"73S* 'GV(Ci
&)|+*+- lu(+"73
∆tn ≤ 2Ai
Sni
(1 + sup
j∈Ni
(∣∣∣[δnij
]p
∣∣∣))
|∂Ci|,
D)&)(; [δnij
]p
=αn
ij
Snij
[λn
ij
]p
(+3Sn
i = maxj∈Ni
Snij,
D)GV$'*vC$'" D8GPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-632|L∞
||W n||L∞ ≤ ||W0||L∞.
<H
W nij =
1
2
((W n
i +W nj
)−αn
ij
Snij
(F (W n
j ) − F (W ni ))· ηij
).
h !
, @$ < B I RI ! / # & ? D< I .-
W ni
W n
j 3 6-!
ηij 3
W nij =
1
2
((W n
i +W nj
)−αn
ij
Snij
A(W n
i ,Wnj ; ηij
) (W n
j −W ni
)).
? 1 "! 2 @ 3# # A
(W n
i ,Wnj ; ηij
) @ /$ &# #/
R #
m- / ([Rn
ij
]p
)
p=1,...,m
A J+ -#! / * ([λn
ij
]p
)
p=1,...,m
3 7 F D
W ni
W n
j
$ &
- # ([Rnij
]p
)
1≤p≤m
3
W ni =
p=m∑
p=1
[ξni ]p
[Rn
ij
]p, W n
j =
p=m∑
p=1
[ξnj
]p
[Rn
ij
]p, W n
ij =
p=m∑
p=1
[ξnij
]p
[Rn
ij
]p,
1 h 2
W n+1i =
p=m∑
p=1
[ξn+1i
]p
[Rn
ij
]p.
, [δnij
]p
=αn
ij
Snij
[λn
ij
]p3,
ξnij =
1
2
(1 +
[δnij
]p
)[ξn
i ]p +1
2
(1 −
[δnij
]p
)[ξn
j ]p.1 h 2
B ∑
j∈Ni
|eij|F (W ni ) · ηij = 0 3
( /:
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
|eij|(F (W n
ij) − F (W ni ))· ηij.
; </ "" 3!( 9 / $ < "# I
A(W n
ij,Wni ; ηij
) 8 W n
ij
W n
i
-! :$ / ηij
K ! :# # =#
3 T &# ?&
3 8 +
A(W n
i ,Wnj ; ηij
) A(W n
ij,Wni ; ηij
) 8& : :- / O ([
λnij
]p
)
p=1,...,m
-#! / # 7 A(W n
ij,Wni ; ηij
)3 /
@ B D&! p -!# # [
λnij
]p
[λn
ij
]p
@ sn
ij
B " ?$ ' , )( D# ? 8- :
[ξn+1i
]p
=
(1 − ∆tn
2Ai
∑
j∈Ni
|eij|[λn
ij
]p
(−1 +
[δnij
]p
))[ξn
i ]p
h
+∆tn
2Ai
∑
j∈Ni
|eij|[λn
ij
]p
(−1 +
[δnij
]p
) [ξnj
]p.
, [γn
ij
]p
=∆tn
2Ai
|eij|∣∣∣∣[λn
ij
]p
∣∣∣∣(−sn
ij +∣∣∣[δnij
]p
∣∣∣)
p = 1, . . . , m
/ "8 0 #L∞ 3
: "
[γn
ij
]p≥ 0
∑
j∈Ni
[γn
ij
]p≤ 1
p = 1, . . . , m.
[γn
ij
]p≥ 0 ⇐⇒
∣∣∣[δnij
]p
∣∣∣ ≥ 1 ⇐⇒ αnij ≥
Snij∣∣∣
[λn
ij
]p
∣∣∣.
H# & 0-! "7 "8$ 0 ?$# ∑
j∈Ni
γnij ≤ 1
H
∑
j∈Ni
γnij =
∆tn
2Ai
∑
j∈Ni
|eij|[λn
ij
]p
(−1 +
[δnij
]p
)
≤ ∆tn
2Ai
Sni
(sn
ij + supj∈Ni
(∣∣∣[δnij
]p
∣∣∣))|∂Ci|.
"( : ∆tn ≤ 2Ai
Sni
(1 + sup
j∈Ni
∣∣∣[δnij
]p
∣∣∣)|∂Ci|
3 D
L∞ #
/ & % ( &# = # /% *)( &# "#$! / " Q :" 9 /- " , D G/I:KLBG :$ =
W nij = 1
2
(W n
i +W nj
)− 1
2sgn
(A(W n
i ,Wnj ; ηij
)) (W n
j −W ni
)
W n+1i = W n
i − ∆tn
Ai
∑j∈Ni
|eij|g(W n
i ,Wnj , S
nij, ηij
)
W 0i = 1
Ai
∫CiW (x, y, 0)dxdy.
1 hrq#2
1 / " D HG/I:KLBG 7 - <
D %I L
2B D 7 O( F </$ "B I ;&#> !/ 9;)( D? "< $#
A(W n
i ,Wnj , ηij
)= A
(W n
i +W nj
2, ηij
).
% ! #" ) ) " $ + 0 "!
K ?# " D 8 / ( "8 " ? 7G/# & - ;= B "4 & > "# ; > B
" J &%$" A $ $ $
:" & ? G/# & : > (R :9 = / !=
3 7 ?9; - 1 & :; - " "# 2
∂W
∂t+∂F (W )
∂x+∂G(W )
∂y= 0
W (x, y, 0) = W0(x, y),
-
W =
h
hu
hv
, F (W ) =
hu
hu2 + 12gh2
huv
, G(W ) =
hv
huv
hv2 + 12gh2
W > ?- @ -!## " -#-
3 h @ & 5 (
3 u v 4 - O- O(R * & ' ' #
?- ! 3 g
: ( " 0 8 J+; "(R" A (W ; η) = A1(W )ηx +A2(W )ηy
- A1(W ) A2(W ) ? < 8 ?= 5 J+0/
3 "
/ :&#
A1(W ) =
0 1 0
−u2 + gh 2u 0
−uv v u
A2(W ) =
0 0 1
−uv v u
−v2 + gh 0 2v
.
< # /-! :$ η =
(ηx
ηy
)3 #
A(W, η) = A1(W )ηx + A2(W )ηy
=
0 ηx ηy
(c2 − u2)ηx − uvηy 2uηx + vηy uηy
−uvηx + (c2 − v2)ηy vηx uηx + 2vηy
,
c =
√gh
$ "" > , & ? B- " <-!# < # $ # H= J+ : #
λ1(W, η) = ~V · η − c, λ2(W, η) = ~V · η, λ3(W, η) = ~V · η + c,
~V · η = u · ηx + v · ηy
8- #
r1(W, η) =
1u− cηx
v − cηy
, r2(W, η) =
0−ηy
ηx
r3(W, η) =
1u+ cηx
v + cηy
.
B " Q H 8 $! /-!
(h, u, v)(x < 6, t = 0) = (5, 0, 0),
(h, u, v)(x > 6, t = 0) = (1, 0, 0). / 0- ? )( # > C- D/< : O(
1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Mesh NS 300x30
0 2 4 6 8 10 12
0
0.5
11
1.5
2
2.5
3
3.5
4
4.5
5
Hauteur de l’eau−Schéma SRNHS
1
1.5
2
2.5
3
3.5
4
4.5
5
h(x,y,t)
x
y
! : ! "# $&%'
3D ( *),+- .0/21 ) -/ . +3456784/∆t = 0.0002
0 2 4 6 8 10 121
1.5
2
2.5
3
3.5
4
4.5
5
9;:=<?>,@BACEDGFHI$JK!E?L)E?MBNPONIQR E4I$JTS RVU,W$X Y FZ[H\LP]$^N R
−3 −2.5 −2 −1.5 −1 −0.5 0−1.5
−1
−0.5
0
0.5
1erreur 2d srpc0.95
9;:=<?>0@&A`_'aDbFHI$JK!EcLed EfJJE4I$JgNFQFH'h4i7jEk']_E4\ R E ' 0.95
_ll
/ > O( / "8/ "(R ?$ = 6-#
∂W
∂t+∂F (W )
∂x+∂G(W )
∂y= 0
W (x, y, 0) = W0(x, y).
-
W =
ρ
ρu
ρv
e
, F (W ) =
ρu
ρu2 + P
ρuv
(e+ P )u
G(W ) =
ρv
ρuv
ρv2 + P
(e + P )v
ρ $ - <
3 V ; <- D& < 0$Q-J 0 "!
7 u
v 3 e
?)( " &#& D# &# B B- 3 P
? Q-&#& ; C &!= # $# " : ( " # &#
P = (γ − 1)
(e− ρ
u2 + v2
2
),
γ -
3# : D# &#& C /"
3T $ < " /
R = Cv − Cp
# : &# A (W,n) = ∇F (W )ηx + ∇G(W )ηy 3
-
∇F (W ) =
0 1 0 0
12[(γ − 3)u2 + (γ − 1)v2] (3 − γ)u (−γ + 1)v (γ − 1)
−uv v u 0
−u[ c2
γ−1+ (1 − γ
2)(u2 + v2)] c2
γ−1− (γ − 3
2)u2 + 1
2v2 (1 − γ)uv γu
∇G(W ) =
0 0 1 0
−uv v u 0
12[(γ − 3)v2 + (γ − 1)u2] (1 − γ)u (3 − γ)v (γ − 1)
−v[ c2
γ−1+ (1 − γ
2)(u2 + v2)] (1 − γ)uv c2
γ−1− (γ − 3
2)v2 + 1
2u2 γv
- c :$ - "# & ? C
3 ( +// H C# 9
c =
√γP
ρ=√γRT .
Z"
/ : :-!# 8 /
λ1(W, η) = V · η − c, λ2(W, η) = V
λ3(W, η) = V · η + c < > 8- &!
R (W ; η) =
1c
1 0 1c
uc− ηx u ηy
uc
+ ηx
vc− ηx v −ηx
vc
+ ηx
|V |22c
− v · η + cγ−1
|V |22
u · ηy − v · ηx|V |22c
+ v · η + cγ−1
, - " : &#
R−1 (W ; η) =1
2
V · η + (γ − 1) |V |22c
−(ηx + (γ − 1)uc) −(ηy + (γ − 1) v
c) γ−1
c
2(1 − (γ − 1) |V |22c2
) (2(γ − 1) uc2
) 2(γ − 1) vc2
−2γ−1c2
2(ηx · v − ηy · u) 2ηy −2ηx 0
−V · η + (γ − 1) |V |22c
(ηx − (γ − 1)uc) ηy − (γ − 1) v
c
γ−1c
B " Q H 8 $!
(ρ, p, u) (x < 6, t = 0) =(12, 106, 0
),
(ρ, p, u) (x > 6, t = 0) =
(1.2, 105, 0
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Mesh NS 150x15
# 1 ' !
0
2
4
6
8
10
12
0 2 4 6 8 10 12
’r11’’r11’
# 8#7 ( V),+$- ++".
100000
200000
300000
400000
500000
600000
700000
800000
900000
1e+06
1.1e+06
0 2 4 6 8 10 12
’p11’’p11’
# b $ $ #78 ( V),+$- ++".
280
300
320
340
360
380
400
420
440
0 2 4 6 8 10 12
’t11’’t11’
# " $ # 456 ( *),+$- ++.
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# XY+Yljqj# # # h WYl+j d ¯X¸ h WV¸[Y+T£j- h Q3:'-jwYl d k TVjji±W@R h ¸iY+j h jT d RIY+W£6WV?W£6¸¯XjsT d d +j- h `?j d WYlWWTVYH|-Yl h WV3Y+|C¸-Q dod X d R6¸ d 3Yl¢?Y;'-Q3jXj d WV tji d H d j-|w+j h Q3jY+ h |-j6RI h+d RWV?j # TV| h Xoj d h |iW d ² d d h+h § cw¨HX¸jiTRRI¸ Y+|C¸-Q d Xjo TQ5jiY76.WYXjo¢?RIj ` c?X)OX2) h X h jo¸-Tj¸oj- +WTVW£Y d HqT d +j-|CWj5# 10¯¹ # [T d WYl+j h«§ [T d ¦)¨ d +WTWVY+¸oT d 6ji|CWV?j3Xj !5+ Y+RTWVl6WV` Q kh YlW § Q k Hµ¨ d +WTVWY+¸sEY+T£j- h Xj f ?j d j-|TVWQ3WV6ji h Xj<RjiH6j«]M# 10¯¹_R hh ¸iY+X h jTVjoR h k TQ'-Q3jcXj«a d TTVµÂ # d 6j h # f #I¹WYl d ji # #¡jiX h § ¹ # 7¨ d RRTVW£¸cTjYl|C¸-Q d Xjc¹ d + %.h WVj-X h W|CYR h ¸i|-¸-X¸qR d h j[R d Y+j[R h ¸-XWV|i+j- h Xjc¹ d + #¡jiX h eR h T d YlWQ°T d 6WV Q3¸ h Wj«X;Yl¢?Y;'-Q3jxXj3 d WVHl tji d H d ji|x+j h Q5j«Yl h |ijxQ°T£6WVXWQ3j-YlWj-TYl h NQ d WTT d jq h6d Rj-LX d Tj # T8) d WXj«X2) ÀYl|C¸-Q d Xj5 TQ3j-Y 6.WVYj- |-jiTTVTjq|-ji h ¸ k.d Yl¸«Y+ h T*) d RR h +aWVQ d +W"X Y+ TVji h Xj f WjiQ d Xj¢?RIj f ?j°Yl hTjiY5WV+j h d |-jiY-`~´# d ^jiE´# deh |-W d $ d deh+h §³k cw¥¥7¨< d k +W . XjiY h ¸iY+T£ d +Yj-|i h+d j d +YRI h TVjoR h k TQ'-Q3joXj« d WVHl tji d H d j-| h ++j-Q3j-HYl h Xj-YQ d WTVT d j-YgYl h |6 h ¸-Y3Xj¢?Rj"¶wjiT d d ¢ # j h Q°Xj § ¯¶w¶ :9 ¨±jxY+j-Y3|- TT dek h+d 6ji h YoR h 7R Y+j-HqjxXWVY+| h ¸i+WY d 6WV X¸-|ij-H h ¸5X;+j h Q3j«Y+ h |ij3WnjiYlq|- Q5R d 6W k Tj d j-|3Tj !.5+Q5¸ h WV?jEX¸i|-j-H h j h RI hod YlY+ h j h T d R h R h W¸i+¸EXj5|-Y+j h d 6WVÀjioH°R h ¸iY+X h jTj[R h k TQ'-Q3jXjo d WV tji d H d ji|q+j h Q3j[Y+ h |ijqYl h OQ d WTT d js"Yl h |6 h ¸ d R h 'iYT*) ¸i+XjcXj-Ys|iXW£6W Y d 5+OTVWQ3WV+j-Y$#´ h T8) d RRTWV| d +WNX^Yl|C¸-Q d f <b d ^R h k TQ'-Q3jXj d WH tj- d H k WXWVQ5jiY+WVaj-T d j-| +RI h6d RWj"W hlh ¸iTWQ' h j"Yl h Q d WVTT d jE h W d T d W h jO Yl h |6 h ¸ `¯ Yjij-|6YwT d R h lj-|6W ¤X¤Yl¢?Yl-'-Q3j]º#V)_±Yl h T d h Q d TVj[Xj-Y<WH6j h d |ij-Yi`I|ij-|iWRIj h Q5jXj h ¸iXW h j TVj-Y°|iQ5RIY d H6j-Y«Xg+j h Q5j Yl h |-j ( «¸-| h W h jT*) ¸i d Rj5R h ¸iXW|6ji h X d Y«jY+jiTj"XW h j-|6W #n´ h T d Y+WQT d +W Q3¸ h Wj"Xj-Y3Yl¢?Y;'-Q3j-Y k W !WXjiYxQ°TV+WXWVQ5jiaY+WVjiTY-`5 +j<T8)·jP+a+j-YlW5X Yl|C¸-Q d % % 0^WH h aXWVR d h [WX d TW d [Q kd h e¹j0a § c¹ 0¯º7¨M`UR deh # d h | |C ej h"§ ~.:9e¨ d R h k T('iQ5j5XWR d Y+WVjE h WXWVQ5jiY+WVaj-TM`.j-¤6WVTWY d HsjqQ5¸6?XjiYYlWQ°TV d ¸-j[ TQ3j-Y36.WVY²¸-T¸iQ5jiH6Y36.WVY<R h RIY+¸ijoR dehcWX d TW d $´ d Y+| d T § c´t7¨ # 0jiT d RIj h Q5j¯Xj h ¸-YlX h j[Tj-Y~R h k T('iQ5jiY|- RT¸iY¯Xj|iHji|6WVjUXj~XWVYlW #µ´ h d W h j~T8)·jP+a+j-YlWoX°Yl|C¸-Q d Xj f ?j d 5+qR h k T('iQ5jiY k W !.WXjiY # Q3WajnY+jiYn|-TVT d k h+d 6ji h YH d la6¸¯j¯j<+aR h j-Y+YlWxXjR h jiY+Y+WV«WH+j h d |-W d TjR hh jiX h joTjcYl¢?Y;'-Q3j°H¢?Rj h+k TWVjojwWVTY±H<WH h aXWVT d Q3¸i+aXjoXjoRj h + h+k.d 6W ¤R dehXj-Y+WV+¸ § 0 h ¥ a`T0¯¶ :9` 0 h a` cºe¨ #n Ylj k.d Y d H5Y+ h |-jil6j¤Q5¸6?XjO d R h j h x|-Hji|i+j- h ¢?RIj hlk TVWj±RIj h Q5j+ d tT d X¸i+j h Q5WV d 6WV°Xj±T d Q d h WV|-j¯YlWj¸i a¸ijoX d Y±T d R d Y+jcR h ¸-XWV|i6ji h XOY+|C¸-Q d f wbw #.´ h T*) d RRTW| d 6WVEX"Yl|C¸-Q d f wb d Yl¢?Yl-'-Q3j k W !.WVXj h W WjiTL`)«| d T|iTj~T d Q d h W|ij~YlWjXj±T*) ¸i d Rj~R h ¸iXW|6ji hR d h T d Q3¸i+?Xj[Xj % # T j-Y k.d Yl¸-jcY+ h T8) d TV h WV6Q5jXjq<j±6a$a|CTV § TVH:9)¨#
vÆH¥
=?> ?D#C " K<G C I $ "%J #& ? /$ FHG "! & I#$ K CMD&%(')#C CMD * J D%G G,+/F - K" KPF F + $ AK"! F./#C IK0%& #C " KAGLC
2
1 32 4 6587:9(;=<>?; A@ )B=C EDF>?G
¹j°R h k T('iQ5jqXjx d WHl tj- d H d j-|°+j h Q3jYl h |-j k WVXWQ3j-Y+W j-TPjiYl h ¸-WR d h TVjY¢aY$;'iQ5jc|-W XjiY+YlY H
h,t + (hu),x + (hv),y = 0
(hu),t + (hu2),x + (huv),y + g(
h2
2
),x
= −gh(Zf),x
(hv),t + (huv),x + (hv2),y + g(
h2
2
),y
= −gh(Zf),y ,
]º#V)_
JIhjiYlT d d 6ji h XjT*) j d `LK
= t (u, v)jiYlwT d aW£6jiY+Y+jji ∇Zf
jiYlwT d Rji+j°X ¼ X]Zf(x, y)
ji ∇Zf
Xj-5+N¼|6W YoRI d H 4i h j-Y° h ¸iTVW(' h j-Yqj-À ¸-¸ h6d T _`|-Q3Q5jYsQ3H h jqT d 6. h jcY+W£ d +jMH
20Q
(Zf )l= 0
Q(Zf )r
= 1Qhr + (Zf)r
= 1.1Q
hl + (Zf )l= 4
Q(u, v)l = (0, 0)
QEÅeY(u, v)r = (0, 0)
QEÅeY
NPORQTS&º#£VU % X"XjcT d°h WV?W(' h j
VI(.)l
YlHcTjiYw d Tji h YXj-Y[ d h W d k TjiYwRH¢?Y+WVj-Y/. d |Cj«Xj«T d XWYl|-H6WVWV+¸°ji(.)rY+HTjiY± d Tj- h Y±XjiY d h W d k TVj-Y¯R¢?YlWj-Y .xX h W£6jXjoT d XWYl|-H6WVWV+¸ #
vÆ
2
´ h T d Y+WVQ°T d 6W ?Q5¸ h Wj5XÀR h k TQ'-Q3j ]º#V)_`PY d TTVY d RRTWj h TVj5Yl|C¸-Q d f wb X¸<6.WX d Y TVj |C d RW£ h j
5# 0Q5Q3j X d YT*) ¸i d RIj R h ¸iXW|6ji h Yj-Y+Y d ¢Y
Xj h j h j d RR h ,+aWQ d +WNXjT d YlT+WNXgR h k T('iQ5j Xj f WVj-Q d d 5+ WV+j h d |-jiY-` Y h j-R h j-YcT d X¸iQ deh |Cj«R h ¸-Ylj-H6¸ijEX d Y § ¥¦a`#¤.¥H)¨L`jc|-Y+WX¸ h Yj R h lji|i6WV¡XÀR h k T('iQ5j3Y+ h T d h Q d TVj3X )·j WH6j h d |ijE Y k 6j-Y d WVY+WtTVjYl¢?Yl-'-Q3j$H
ht + (huη),η = 0
(huη)t +(hu2
η + g h2
2
)η
= −gh(Zf),η
(huτ ),t + (huηuτ ),η = 0,
]º# _
d j-|uη =
K · η, uτ =K · τ ` η ji
τYlH h jiY+RIj-|i+WV j-Q3j-HcT d h Q d TjqjiwT d d jiH6jM.
T*) WH6j h d |ijeij
`ji(.),η
j-YlT d X¸ h W£¸ijqTVj[T«X" j-|6j- h h Q d Tη#
¶ d Y|-jc| d YsT*) ¸i d Rj[R h ¸-XWV|i+j- hUn
ij
X"Y+|C¸-Q d f wbwOYA)·¸i| h WVXjcT d Q d ' h jcY+W£ d H6j
Unij =
1
2
(Un
i + Unj
)− 1
2sgn
(∇Fη
(U)) (
Unj − Un
i
)
+1
2
∣∣∣∇Fη
(U)−1∣∣∣Qn
ij
]º# ¦_d j-|
U =
h
huη
huτ
, Fη(U) =
h
hu2η + g h2
2
huηuτ
, Qn
ij = −g2
(hi + hj)(Zf j
− Zf i
)
0
1
0
]º#·ÆH_jiU
jiYlT*) ¸i d sXj f ?j[X¸cR d h
U =1
2(hi + hj)
1
ui
√hi+uj
√hj
√hi+
√hj
ηx +vi
√hi+vj
√hj
√hi+
√hj
ηy
−ui
√hi+uj
√hj
√hi+
√hj
ηy +vi
√hi+vj
√hj
√hi+
√hj
ηx
.
]º# _
¹j-Y± d Tji h YsR h R h jiYsY+HX¸-j-YR d hλ1 = uη, λ2 = uη +
√gh
jλ3 = uη −
√gh,
]º#·ºH_T d Q d h W|ijXj-Y± j-|i+j- h YwR h R h j-YsjisT d Q d h WV|-jXjiY±j-|6ji h Y<R h R h j-YsWV j h Y+jcY+HX?¸-jiY<R d h
R(U) =
0 1 1
0 λ2 λ3
1 uτ uτ
, R−1(U) =
−uτ 0 1
−λ3
2√
gh1
2√
gh0
λ2
2√
gh− 1
2√
gh0
,
]º# _
vÆ
sgn(∇Fη
(U))
= R(U) sgn(Λ(U)
)R−1(U)
ji ∣∣∣∇Fη
(U)−1∣∣∣ = R(U)
∣∣∣(Λ(U)
)−1∣∣∣R−1(U).
n16. "¸i| h WVsT*) ¸i d RIj[|- hlh j-|6ji h XjqT d Q d W(' h j[Y+WV d H6j
W n+1i = W n
i − ∆tn
Ai
∑
j∈Ni
g(W n
i ,Wnj , Q
nij, η)
+ ∆tQni ,
]º# 9_d j-|
g(W n
i ,Wnj , Q
nij, η)
= F(W n
ij
)· ηij, W n
ij =
hnij
(huη)n
ijηx − (huτ )
nij ηy
(huτ )nij ηy + (huτ )
nij ηx
.
]º# _
ji
Qni = −g hi
Ai
0∑
j∈Ni
Zij · (ηij)x|eij|∑
j∈Ni
Zij · (ηij)y|eij|
,
d j-|Zij =
ZiAi + ZjAj
Ai + Aj
1 > @ 5?; A@ B 9 9 @ 658<CA7
X¸A6.WV°À j-T¯jiY+R d |-jX2) d RR h ,+WVQ d 6W `JIÀT8)· d RR h a|CjET*) ¸i d RH¢?Y+WVjVR d h jXWVYl h W k 6WV5TVW¸ d W h jsR d h Q3 h |-j d 5+`W@jiYl±T8)·jiY+R d |-jXjiY¼|6WVY d j-Y±R deh
|-jiTTTj # d RRTVWjtj~X¸-Q d h |CjW d ¸6¸a6WTVWYl¸-jtR d h #nTVQ d W § nTQ5)¨L`7|-j++jX¸iQ d h |CjYljX¸-|iQ3RYlj[j-OXji5+O¸i d RIj-Y Hjq¸ d RIj[Xj h j-|- Yl h |6W "Xj-Ys¸i d 6Y± d |Cj[jisX h W£6j `jijc¸ d RIjcXjcTWQ3WV d 6WV3Xj-Ys h+d XWjiH6YsR h ¸i?WV+j h TjiY±Y+|iWTVT d +WY #À ] h ji|-Y h |i+WOTWV¸ d W h jwXj-Ys¸i d 6Y6_
RIWHXij
Xj°T8) deh 4i+jΓij
Y+¸-R d h6d HcXji5+ TQ5jiYXj«|- h TVjCi
jCj
`@ X¸<6.W£Xj- +¤ d TVj- h Y<Xj
V`&j .5 d |Cj
V −ij
X d YwT d |-j-TVTTVjCi
`&ji<j .3X h WV+jV +
ij
X d YwT d|-jiTTTj
Cj
R d h H
V −ij = Vi +
1
2~∇Vi
~GiGj
V +ij = Vj −
1
2~∇Vj
~GiGj.
]Mº #V)¥_
Gi
jGj
YlH h j-YlRji|i6W£jiQ5jiETjiY k.d h ¢?|ij-H h j-YEXj-Yx h W d TVj-YCi
jCj
` ~∇Vi
j ~∇Vjh jiR h ¸iY+j-H+j-HTVj-Ys h6d XWji+Y±Y+ hCi
jiCj
#¹ d ¼ |i+WOXj !.5+ h j-Y6jqT d Q4-Q3j `Ylj-TVYsY+j-Y d h Q3j-H6Ys|C d j-Hv`.j d H
∫
Γij
F(V,−→n ) dσ = φ(V −ij , V
+ij ,
−→ηij)mes(Γij).]Mº #V)_
vÆH¦
2
V +ij
Gj
Γij
Xij
Gi
V −ij
NPORQTSIº#,U¹j R h k T('iQ5j | h |iW d TX d Y«|-jil6j¸ d RIj j-YxT*) ¸i d TV d 6WV;Xj-Y« h+d XWVj-H6Y°Xj T d YlTa6W Y+ h |C d j3TVQ3j3Xj5|iH h Tj # 0ji|-WRjiL4i h j5j@ji|i+¸EXj3XW£@¸ h j-H+j-YqQ d W(' h j-Yi`j-R d h 6W|iTWVj h YlWV d HTj[¢?RIjoXjq|-jiTTVTjiY±6WVTWYl¸-jiY$# "R h ¸iY+j-H+j°j[6ji|CWV?jXjq| d TV|-TXj h6d XWj-H+YX d YtTjs| d Y~JI336WVTWVY+j¯XjiY~Q d WVTT d j-Y h W d T d W h j-Y3? d X h6d T d W h j-Y$# "! $# !%!&´ h " h W d Tj
Ci
`|iY+WVX¸ h YT*) j<+aR h j-Y+YlW
∆(Vi, Vj) = Vi + (xj − xi)∂Vi
∂x+ (yj − yi)
∂Vi
∂y− Vj
JI(xi, yi)
j(xj, yj)
Y+H h jiY+RIj-|6WV j-Q3j-HcTjiYw|ia h X¸-j-Y[Xj-Y|ij-H h j-YcXj° h+d aW£6¸oXjCi
jiCj '¹js+j h Q5j
∆(Vi, Vj)h jiR h ¸iY+ji+j<WV|-WT d XWV¸ h j-|ijsj-H h jT d d Tji h jij-|6WV jsXj
Vd x|-j-H h j
Xj3T d |-jiTTTjCj
jT d d Tj- h k 6jij d N|ij-H h jEXjCj
R deh X¸ijiTRRjiQ5jioTWVQ5W£6¸oXjT d ¼|6WV
VX¸A6.WVj d |-j-H h j"Xj
Ci '¹jO h6d XWj-HxYl h T d |-jiTTTj
Ci
jiYl5¸i d T¸j-Q5WVWQ3WY d HT d ¼|i+W d X h6d +WjMH
Ψi(∂Vi
∂x,∂Vi
∂y) =
∑
j∈N(i)
|∆(Vi, Vj)|2
JI ∂Vi
∂x
j ∂Vi
∂y
YlH d T h YYlTa6W YX Y¢?Yl;'iQ5jOTVW¸ d W h j5XjEXj-5+À¸- d +WYE. Xj- +W|ijiY<YlWV d H H
∂Ψi(X, Y )
∂X= 0
∂Ψi(X, Y )
∂Y= 0.
vÆÆ
<"| d TV|-T@Y+WQ3RTVjXj∂Vi
∂x=JxIyy − JyIxy
D
∂Vi
∂y=JyIxx − JxIyx
Dd j-|qTjiYs d 6W Y
Ixx =∑
j∈N(i)
(xj − xi)2 , Iyy =
∑
j∈N(i)
(yj − yi)2
Ixy = Iyx =∑
j∈N(i)
(xj − xi)(yj − yi) , D = IxxIyy − I2xy
Jx =∑
j∈N(i)
(xj − xi)(Vj − Vi) , Jy =∑
j∈N(i)
(yj − yi)(Wj −Wi)
Z ]¼TWVQ5W£ d +WxXjiY<RIj-H6jiYC_'¹ ) d RR h ,+aWQ d 6WVO k +j-j d ji|°TjiYw¸ d +Yw¸i d TV¸-YX d YT*) ¸i d Rj
1`&RIj- 4i h jXWY+RIj h Y+W£j
jioR h ?XW h jxXj3 j d +¡jP+? h jiQ d j-H h j5TVj-Yq|-jiTTTj-Y'´ hd +6¸ij h |-j-YqYl|-WTVT d +WY-`
6WVTWVY+j° Yl|C¸-Q d Xjx¢?RIjTV D
k 6jiNjiN+WTVWY d wXj-Y[ d TVj- h Y[TWQ3WV+¸-jiY ∂limVi
∂xji ∂limVi
∂y
d TWVj- Xj ∂Vi
∂x
ji ∂Vi
∂y
`@X d Y5]Mº')¥H_
' R h ¸iY+ji+j5WV|-W ¢aRIjxXj«TWVQ5W£6ji h Y
WRji74i h jc6WVTWVY+¸X d YsTj[| d Y k WXWVQ5jiY+WVjiT@RI h XjiY<Q d WTVT d j-Y"Yl h |i+ h ¸iY' 8K!$ !
a h |C d jTVQ5jsXj|- h TVjCi
`HTjiY~ h+d XWVj-H6YtY+ TWVQ5W£6¸-YXjwT d Q d W(' h jsY+W£ d +j H
∂limVi
∂x=
1
2
[minj∈N(i)
sgn
(∂Vj
∂x
)+ max
j∈N(i)sgn
(∂Vj
∂x
)]min
j∈N(i)
∣∣∣∣∂Vj
∂x
∣∣∣∣ .]Mº' _
¶wjQ4-Q3jcR h ∂limWi
∂y'0j++jTWVQ5W£ d +W5Rji 4i h jo¸6jiXjoY+ h T*) j-Ylj-Q k TVj
N(i)Xj-Y
h W d Tj-Y d ¢ d Hw¤YlQ5Q3ji¯|-Q3Q° d ji|Ci '
1 <<VB=;=7FC>?; @ D&9 ;
d RRTVWjsTjY+|C¸-Q d f wbxR h TVj-Y~| d Y| h WV6WVj-Y6j-Y6¸<Y+ h TjsR h k T('iQ5jXj[ d WHl tj- d d ji|q+RI h6d RWj[W h+h ¸-TVW(' h jwWXWVQ5jiY+WVjiT@X d YsTj[|C d RWV h j
4 'n2D
TVj¼X"j-Y<X¸A6.WXjqT d Q d W(' h jY+WV d H6j
Z(x, y) =
0Y+W
(x, y) ∈ [−10, 0] × [0, 1]
1Y+W
(x, y) ∈ [0, 10] × [0, 1].
vÆ
V
2
K# !% &FK
¹j-Ys|-XWV+WY±WVWV+W d TVj-Y~X d Ys|-jc| d YsY+HX¸-j-YR d h
(hl, ul, vl) = (5, 0, 0) (x, y) ∈ [0, 6] × [0, 1]
(hr, ur, vr) = (1, 0, 0) (x, y) ∈ [6, 0] × [0, 1]
0j| d Yj-Yl¼ h Q3¸ER deh j Xj Xj h+d h ¸ d |i+W@`jXWYl|-H+WWV+¸EXjE|-H d |i«j«|C?|
'¹ d |-Q3R d h+d WYl¡Xj5T d Y+TV6WVNQ3¸ h Wj d j-|T d YlT+W;6¸i h WVjEQ3H h jk Wji¤jc|ijqYl|C¸-Q d | d R+j k Wji¤TVj[|Ca|
'
! # !% &&K
¹j-Ys|-XWV+WY±WVWV+W d TVj-Y~X d Ys|-jc| d YsY+HX¸-j-YR d h
(hl, ul, vl) = (5,−4, 0) (x, y) ∈ [−10, 0] × [0, 1]
(hr, ur, vr) = (1, 9, 0) (x, y) ∈ [0, 10] × [0, 1]
¹ d |-Q3R d h+d WYlÀXjiY h ¸-YlTV d 6Y«Q3¸ h WjiYE.t = 0.7s
d j-|TjiY h ¸iY+T£ d +Y°6¸i h WVj-YR h |-jcR h k T('iQ5jwR h ajcjcTj[Y+|C¸-Q d f wbw dxk Wj-O| d R6¸cT d R d YljqXj h6d Y+W£6W ]¼ W h 6. h jiY NPORQTS&º
'a` NPORQTSIº
'xjiVNPORQTSIº
')¦H_
'¹j-Y|i hlk jiYXjiYWYl) d Tj- h YtY+H~Xj-Y~X h WV6jiY~R d h d W£6jiY|ijwW&|- 16 h Q3j?jwTjsR h k TQ'-Q3j|-Y+j h j°Yl | d h6d |i;' h j
1D]¼ W h 6. h j NPORQTS@º
'º` NPORQTSº
')¥xji NPORQTS@º
')¦H_
'nY+WV6jo
R h jiXOO| d Y±RTY h+d WXjXHsT d d +j- h .° d |Cj[j-Y40
¼WVY¯RTVY¯ h+d Xj[j[|ij-TTVjO.X h WV6je`TjiY±X¸-j-YWWV6W d TjiYX d Ys|-jc| d YsYlHwX ¸-jiY<R d h
(hl, ul, vl) = (4, 0, 0) (x, y) ∈ [−10, 0] × [0, 1]
(hr, ur, vr) = (0.1, 0, 0) (x, y) ∈ [0, 10] × [0, 1]. h j-Q d h j[jcRI h |ij[| d Ys¸i d TjiQ5jiH±TjYl|C¸-Q d f wb | d T|iTj k Wj-OT d R d YljcXj h+d Y+W£6WVÀji d k Wj- | d R+¸TjE|Ca|;]uW h 6. h j NPORQTSnº
' ` N ORQTStº
'¦¦ j NPORQTSnº
'¦ _
'¶wj RTVYqTj3R d Y+Y d j?.¤T*) h X h j3Xj- +ÀjiÀj-YlR d |-j d j-| T d +j-|CWjO# w 0¯¹Rj h Q3jioXjh ¸iY+X h j«TjoR h k T('iQ5jqXj°T d XWYl|-H+WWV+¸oY d +W d W h j ` h ¸iY+T£ d H[Xj°T d X k TVjYlW?T d h WV+¸ R h ) j- d H . T d ¼WY«XjiYx|-XWV6WVY«WVWV+W d TVj-YXj-Yx deh W dek TjiY°RH¢?Y+WjiYEj5Xj¤T dXWYl|-H+WWV+¸[XO¼X
'
vÆHº
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
"!#$%%'&)(% *+F-,
−10−5
05
10 0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Hauterde l’eau−Schéma SRNHS
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x
y
h(x,y,t)
.0/1324)5-67##89;:)"<*-#=>5@?A>#CBDFEG*IH %)JLK F@MNO)PnFQ<R O*+OAAS
∆t = 0.0001 TBUV % *-,W<
5+"*-OF
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x
h(x
,y0
,t)
Hauteur de l’eau−SRNHS
AnalytiqueSRNHS (sans MUSCL)
NPORQTSº'$UYX4Z[>\+]_^`^a9bdc)b[e
f ^W[>\hgiZ[>j_^lk=mon fiprq>sut'vxw Zy)`;`z v egWZy f zyA[|
∆t = 0.0001~ q z f- \b'e
f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration de la hauteur
NPORQTSº'º U3^ v z v Z'b^=[>\ v `^C b
c)b[ f ^=[>\
vÆ
V
2
−10 −8 −6 −4 −2 0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4Champ de la vitesse de l’eau
NPORQTSº'VU X4c)b jC`^ b Az f ^ v+v ^
−10−5
05
10
0
0.2
0.4
0.6
0.8
1−4
−2
0
2
4
6
8
10
Vitesse de l’eau−Schéma SRNHS
−2
0
2
4
6
8
x
y
u(x,y,t)
NPORQTSaº'9VU X4Z[>\+]7^.`^. b Az f ^ v+v ^
A[>^.m n fIprq)sLt"v w Zy)``>z v giZy f zyA[|
∆t = 0.0001 q z f- \+b f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 10−4
−2
0
2
4
6
8
10
x
u(x
,y0
,t)
Vitesse de l’eau−SRNHS(sans MUSCL)
AnalytiqueSRNHS (sans MUSCL)
! #"%$'&(*)()+, -.0/21435&7689':;)=<>-?@AB$!) , ?BC&($ ?%ED
∆t = 0.0001DF-GH-8I$ &JKL MC
&($ ?@)
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration de la vitesse
NPORQTS&º')¥,U ^ v z v Z'b^=[>\ v `^ b
Az f ^ v+v ^
vÆ 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o7A "!#$%%'&)(% *+F-,
−10−5
05
10
00.2
0.40.6
0.81
−20
−15
−10
−5
0
5
10
Schéma SRNHS−Débit de l’eau
−15
−10
−5
0
5
x
y
q(x,y,t)
; 24>5+6_#P# ,=6>0*?A># BD E *IH %>JuK F9MNO_nF R O*+OAAS
∆t = 0.0001 TBUV % *-,=5+"*-OF
−10 −8 −6 −4 −2 0 2 4 6 8 10−20
−15
−10
−5
0
5
10
x
q(x
,y0
,t)
Débit de l’eau − SRNHS (sans MUSCL)
AnalytiqueSRNHS (sans MUSCL)
NPORQTS<º')¦ U X4Z[)\-]7^ `[o` ]>z f
gWZ[>j7^ k~m n fIprq>sut'v@w Zy)` `z v giZyef zyA[|h
∆t = 0.00019 q z f- \b'e
f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration
NPORQTSµº'vÆPU;^ v z v Z''b"^W[>\ v `[@` e
]>z f
vÆH
V
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Mesh NS 300x30
NPORQTSº'µ U7ZAZ `[ bz b^
[ f zz v qq q
0 2 4 6 8 10 12 0
0.2
0.4
0.6
0.8
1
1
1.5
2
2.5
3
3.5
4
4.5
5
Hauteur de l’eau−Schéma SRNHS
1
1.5
2
2.5
3
3.5
4
4.5
5
x
y
h(x,y,t)
NPORQTSaº'vºVU X4Z[>\+]_^r`^.a9b@c)b"[e
f ^=[>\CA[>^ m n fiprq>s"vw Zy)` `z v egiZy f zyA[|
∆t = 0.00034k= q z f- e
\+b f zZy v
0 2 4 6 8 10 121
1.5
2
2.5
3
3.5
4
4.5
5Hateur de l’eau−Schéma SRNHS
SRNHS (avec MUSCL)Analytique
x
h(x,y0,t)
NPORQTSº' U X4Z[)\-]7^.`^ra9b c)b[e
f ^W[>\hgiZ[>j_^lk=mon fiprq>s"vxw Zy)`;`z v egWZy f zyA[|
∆t = 0.0003k~ q z f- \b'e
f zZy v
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration
NPORQTS&º' 9,U ^ v z v Z'b^=[>\ v `^ b
c)b[ f ^=[>\
µ ¥
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Mesh NS 300x30
NPORQTSº')MU7ZAZ `[ bz b^
[ f zz v qq q
0 2 4 6 8 10 120
0.5
1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Vitesse de l’eau−Schéma SRNHS
0
0.5
1
1.5
2
2.5
3
x
y
u(x,y,t)
NPORQTS º' ¥ U X4Z[)\-]7^ `^ b
Az f ^ v+v ^3A[>^ m n fIprq)s v w Zy)``z v gWZy f zyA[|
∆t = 0.0003'k~ q z f- e
\+b f zZy v
0 2 4 6 8 10 12−0.5
0
0.5
1
1.5
2
2.5
3
3.5Vitesse de l’eau −Schéma SRNHS (avec MUSCL)
SRNHS (avec MUSCL)Analytique
x
u(x,y,t)
! #"%$'&)(* ,+-+.0/123547698 *;:=<?> @ +BACD?E&#+.(/FGD * &#DHBI
∆t = 0.0003I47JK < & *L M"N(
* &)D?+
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration
O#PQR #,+0&+N$S"G ),M+5 "$H& * ,+-+.
TU
V
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Zoom du maillage 150x15
NPORQTSº' ¦MU7ZAZ `[ bz b^
[ f zz v k "q k
02
46
810
12
0
0.2
0.4
0.6
0.8
1−1
0
1
2
3
4
5
6
7
8
Débit de l’eau−Schéma SRNHS
0
1
2
3
4
5
6
7
x
y
q(x,y,t)
NPORQTS<º'7Æ U X4Z[>\+]_^P`^ ` ]>z f
A[>^ m3n fIprq)s v4w Zy)`G`z v gWZy f zyA[
0 2 4 6 8 10 12−1
0
1
2
3
4
5
6
7
8Débit de l’eau− Schéma SRNHS
analytiquueSRNHS (avec MUSCL)
x
q(x,y,t)
NPORQTS<º' U X4Z[)\-]7^ `[o` ]>z f
gWZ[>j7^ k~m n fIprq>s"v@w Zy)` `z v giZyef zyA[|
∆t = 0.0003 k= q z f- \b'e
f zZy v
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration
NPORQTSµº' º U;^ v z v Z''b"^W[>\ v `[@` e
]>z f
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o7A "!#*+nF , $%%'&)(%
−10−5
05
10 0
0.5
1
0
0.5
1
1.5
2
2.5
3
3.5
4
Hauteur de l’eau −Schéma SRNHS
0.5
1
1.5
2
2.5
3
3.5
x
y
h(x,y,t)
24>5-67#F#F8 :)">*-#W)5 ?A># BD E *IH JL( FMNO)nF R O*-OA
T ∆t = 0.0002 T $BV % *-,W<
5+"*-OF
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
3.5
4Hauteur de l’eau− SRNHS (MUSCL)
x
h(x
,t)
SRNHS (MUSCL)Analytique
NPORQTSº'VU X4Z[)\-]7^.`^ra9b c)b[e
f ^W[>\hgiZ[>j_^lk=mon fip k s vxw Zy)`;`z v egWZy f zyA[|
∆t = 0.0002 q z f- \b'e
f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration de h
NPORQTSHº'¦¥ U a9^ v z v Z''b"^W[)\ v `^ b
c)b[ f ^=[>\
¦
V
2
−10 −8 −6 −4 −2 0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4Champ de la vitesse de l’eau
NPORQTSnº'¦ U X4c)b j v `^Cb ze
f ^ v+v ^
−10 −5 0 5 100
0.5
1−1
0
1
2
3
4
5
6
Vitesse de l’eau−Schéma SRNHS
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x
y
u(x,y,t)
NPORQTS±º'¦ UX4Z[>\+]7^ `^ b Az0e
f ^ v-v ^CA[>^dm n fIp k s v w Zy)` `z v egiZy f zyA[|
∆t = 0.0002 q z f- e
\+b f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 10−1
0
1
2
3
4
5
6Vitesse de l’eau− SRNHS(MUSCL)
SRNHS (MUSCL)Analytique
NPORQTSsº'¦ ¦ UX4Z[>\-]7^ `^ b ze
f ^ v+v ^hgiZ[>j_^ k~m n fIp k s v|w Zy)`l`z v egWZy f zyA[|
∆t = 0.0002 q z f- \b'e
f zZy v
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration de la vitesse
NPORQTS&º'¦ ÆU ^ v z v Z'b^=[>\ v `^ b
Az f ^ v+v ^
eÆ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o7A "!#$%%'&)(% *+F-,
−10 −5 0 5 100
0.5
1
−1
0
1
2
3
4
5
6Débit de l’eau−SRPCS (MUSCL)
x
q(x
,y,t)
y
0
1
2
3
4
5
24>5+6_# ,=6>0*?A># BD3E*IH J ( F4MNO)GnF R O*-OA
−10 −8 −6 −4 −2 0 2 4 6 8 10−1
0
1
2
3
4
5
6
x
q(x
,y0
,t)
Débit de l’eau−SRNHS (MUSCL)
SRNHS (MUSCL)Analytique
-@0B BJK$ &, -.0/21435&76#/-9 )=<>-?@AB$!) , ?BC&($ ?%
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Iso−Concentration du Débit
NPORQTSµº'¦:9 U;^ v z v Z''b"^W[>\ v `[@` e
]>z f
2
=?> ?D#C " K<G C I $ "%J #& /$ FHG "! & IK0% L" K $ #C IK & #C " K<G C
2
1 32 6B 4 ; <VGCA58; !"#$#%&'#%()1''*,+-./#0) '')(1-243 .5*#%6'#%./.738*9':-*7; +<9'243-,#%('+=)#47)'35+>)9'*@?%'A/B6'#%('*?,,"#%C)DBE.F;$./#HG%9'&))'3 +JI-3)'DE243-!"*K9243-,"#%D243L)9'*;?%AE<'#*;?,,"#%M)KB>243-,49K).N#3(?%.N4.N#HG%9
',L'7./#0) ''O' '*;.N)7);-,#%E#%AL(9':%:%'P
'QR5ST#*./L:%9'9*"L)U'.N#0) ';V9?#W2439O)-,YXZL[U\ 9 ]'^ _#Y-,*
∂t
/)9*@?%9'O-*V*"(`#%*,3E".N('+ ∇· *)Z#%a9'*"!"'3(*V)(YBb)@?%'*;:%'('Y∇ *)Z#%`9*",'3*)(36:*")'A
'V'U''9243-,"#%)U'#*;?,,"#%& )Z9'*?'AH• '9203,"#%)U'#%('*?,,"#%&)UB7.N; H
∂t(αkρk) + ∇ · (αkρkUk) = 0, ced ' fAg
• '9203,"#%)U'#%('*?,,"#%&)UB7243-A"@"9U)U./#%3(?%.NAH
∂t(αkρkUk) + ∇ · (αkρkUk ⊗ Uk) + αk∇Pk + (Pk − P ik)∇αk = αkρk~g, ced ' 'h4g
#JIk
5*a#%*;535+_1-,'U:A,i''3( c ?,a'3(* g (k = v)L243)
(k = l) ' αk
;Uj3 +F)(8?0) 203(a?9'* 6-
αv +αl = 1+ρk
;kO)@"9,+Uk = t (uk, vk)
;U?0@"';Pk
; B7*;';#%'
1 5 B A;=5l nm 9 > 9 o 3b@?% ,35)3b;G0;-''./+ c 9243-,"#% ced ' HfAg + cpd ' h4g;g 243*)Z#%59'*;q@rs)(';3'+WG73()2)Z#%43'R243t)2)Z9'243-,#%u+ 243q*)Z#%L'#) ''*;q'#%./.Nv)'R('#%43w*'-' XZxy#%3 :9 ] H• "35+$)U*9'(' H
αk k = v, l• (*';#%.N#HG'H
Pk k = v, l• ?0@"'; ./#HG%'(',H
uk k = v, l '^ C'3(*5;#W$'Y?,*;BP('Y35+WB*;3?,A''+v*'9'53 +C*;9''9)'A'/-*7)#%k#%36)'"Pk"1*./#W)WGW.N243' H• .N;'?#%3./243.N#HG'H
ρk k = v, l 'Qw'V3(*'"*./')?0*#%A 4"*;U<+W*.N9V3&.N#HG%E)(*;',"#%D)LST'*;.N"3*; H• (*';#%.N#HG'8,35+&A'*;Se,'' H
P ik k = v, l0y'w#%3*. 'O.7j1#*y)*',"#%z)8ST'*;.N"3*;L*#P-P'./'Ak3k./a#%*r
jA'V3,*)3 #%'*A L.N#0) 'L)1-;243'7.5)35+DI-3)''
d
" "
1 B=C>?; @ m 9 > 9 B=CE<9 5?58; @ 7L> C @ > 5
V'O, ,#%F#%3'#)9'*;#%V#%y)2) 9j,;3?,A',HP = Avρv
γ ,H?'
γ = 1.4
Av = 105 'ρl = klP
a, a = 4.73 × 10−5 kl = 987.57 '0y;"U*B,#%D#%'*U7*#%6A"*;SeB3&@?% ,36)(8) A"'*Se'8:Air 203()H
Pl = Pv = P iv = P et Pl − P i
l = ρlΘ, ced ' g
Θy3/'#0 'Az1#%./#%::'VH?' B?0";V' 243 8) u )BL);`*;#%N3/? 3
)k8) 4'*Se'z';=*v'5#%./("'
#3*=3v)y)9" ;3*=B *#%5)kVI-3("3,"#%#%6`3(V'#%;3'*'*9ST9'*' ;3?,A'OX xy9* d + QR1 4+ Q :9 ] 'W'#%<8) , ,#%_)3<;j19'.N(\ 3 *#%P((''./bPI3)b./#%#0)./'(#'v#%`3(q9'**
Θ = δθ+=#%5(243zzj19.F(\Y35GW;'.N#%*;:%'A=8) ;#%3(
)UB5*;';#%6A'*;Se,'B,8Ea#%Aδ = 0
#%*;k)36 '3w)U7.F,*';:% ?%O.N9"1#0)L) %
'o #%3(:%'
'
=?> ! I $ J D%K<G C & D #& D%K $ I $ & GI#!T IK - /" K' $
Qw./#0) '')(1-243/.b8) 9243P*;8 )Z9'*@ )(7.F,('*L3(?,A"MH
∂tW + ∂xF (W ) + ∂yG(W ) + S1(x, y,W ) = S(W ) cpd ' d g
W'5b?%"3*7)Y?,*P'U'#%('*?,,"@STY('#%43+
FG
ST#%("#%O)NI-3 +1AG02438)8O-*"V#%;'*;?,,?+
S1
';yBU-*"V#%F'#%;'*?!"?8S
y"*./#%3(*'#4V)9<6-'-*
W =
αvρv
αvρvuv
αvρvvv
αlρl
αlρlul
αlρlvl
, S(W ) =
0
αvρvgx
αvρvgy
0
αlρlgx
αlρlgy
S1(W ) =
0
αv∂P∂x
αv∂P∂y
0
αl∂P∂x
+ (P − P il )
∂αl
∂x
αl∂P∂y
+ (P − P il )
∂αl
∂y
,
" A
?%'Uk = t(uk, vk)
~g = t(gx, gy)
F (W ) =
αvρvuv
αvρvu2v
αvρvuvvv
αlρlul
αlρlu2l
αlρlulvl
, G(W ) =
αvρvvv
αvρvuvvv
αvρlv2v
αlρlvl
αlρlvlul
αlρlv2l
.
0y#%./("L"43 )'V#k)2)Z9", '#%;)9*9''+a#%6`3(V)*;U203(O/1U?,`3*'8)#'#%()(9'*9O#%./.NU36:AiU-*Se9?%#%3-A )Se'#%6;'A"*;#%243U8b(1-243)' Se,P'./'Av#%./*P,+,?#%*;#%.N(*';P8X xy#3 :9(+%[UQ 0 ']
'Qw'qP+((*';#%
)'V)9*?9'Pw1
Pw4
+- ?%w1 = αvρv
w4 = αlρl
#%AV)#%(9' -,*,HPw1
=γP
ρvαv
, Pw4=
γP
ρlαvV)9'*;?9''V)(αv)w1
(αv)w4
#%AV)#%(9' -,*(αv)w1
= 0, (αv)w4= −αv
ρl
.
V'U'S1(x, y,W ) = A(W )
∂W
∂x+B(w)
∂W
∂y#JI6'.N,*'A(W )
B(W )
#%AV)#%(9' -,* H
A(W ) =
0 0 0 0 0 0
αvPw10 0 αvPw4
0 0
0 0 0 0 0 0
0 0 0 0 0 0
αlPw10 0 αlPw4
+ Θ 0 0
0 0 0 0 0 0
,
B(W ) =
0 0 0 0 0 0
0 0 0 0 0 0
αvPw10 0 αvPw4
0 0
0 0 0 0 0 0
0 0 0 0 0 0
αlPw10 0 αlPw4
+ Θ 0 0
.
9
" "
=?> "%J FHK D%K<GLC I & D GI#
#%3(* Se*LBb.73,"#%D43.N9*203(O)'O*#P('.NU#%3(8(203('*#j19.F/)
?%#%3./'36-(\D)9A6)-,Lj1-("*;5)UB5.F,('*;3?,A"MH
#%3#%3("'*/./91#W)(Y)O:bK)'35+_9"a''+a)-(8B/*.NQ''*;O9ja#%63(";85*"(;ST#%*;.F,#%F)2)q3'*'6A':%*,"#%$'6".N
∂W
∂t= S(W )
W (x, tn) = W n(x)
cpd ' g
?%OL#%3(? 3Ej19.F (\D#%&*9#%3()_8;G0;-''./O;3?,A,H∂tW + ∂xF (W ) + ∂yF (W ) + A(W )∂xW +B(W )∂yW = 0
W (x, tn) = W n+1(x).cpd ' 9 g
Qw'3w)36:%L)B7.N,"*;L)-B51U*;9')"3*UST'*/,#%*;'+;#%-,*V3(.N9"1#0) "9*",? 3(9 *
'o #%3:'yP93*z*) :#%*@"1./k) V#% Wj143("i
X o # 9W+ o xk + ] + #%@ * 5.N9"1#0))Oa'*"3*;P-,#%6-*)@"9A"*;#W)3("U-,*' #3.NuV #%P`#*",'3*;X + 0y#%* 0+0k :9(+0y#%* ]
'#%3(*V9'*;*U8) 9jaL*9)"3*#%&(9 *;8L;G0;''./ cpd ' g )(5.F((''*;3(?,A"
∂tW +M(W )∂xW +N(W )∂yW = 0 ced ' g ?%'M(W ) = ∇F (W ) + A(W )
N(W ) = ∇G(W ) +B(W ) 'QR%,'#%P'()(#%A #E*;#%a#%L)U'3*L:%L;
J (W, η) = M(W )ηx +N(W )ηy. cpd ' g
1 32 C DF>?G ; >(D&9(C>(; z ,y#JINBL-*".F:-*k)('t?,'3(* *#%(*'z)VB '#%P('( J (W, η)
;`"@")?,ABY-,*;"8*;9'',+W#%D9'*;8) 9ja*;9')"3*k)3&;j19'.N (\Fa#%3*k*#P('.NL)U7.F((''*;L3@?,A" H
W nij =
1
2
(W n
i +W nj
)− 1
2sgn
(J(W , η
)) (W n
j −W ni
) cpd ' Hg ?%'
W';z*)Z9",z.N#HG%F)8\V#0V '39VA"* z'3
Ci
Cj '
z'8' *)Z9j!.N#HG'6)\V#0bX 0y#* ] OX #%3 d ] `#3*?,*;BP('y1AGW;2438'V)#%9U-,*
1
αl
=1
2
(1
(αl)L
+1
(αl)R
), αv = 1 − αl, cpd ' g
p =(αlp)L + (αlp)R
(αl)R + (αl)R
, cpd ' f%g
" V
uk =
(√αkρkuk
)i+(√
αkρkuk
)j(√
αkρk
)i+(√
αkρk
)j
; k = l, v, ced ' h4g
vk =
(√αkρkvk
)i+(√
αkρkvk
)j(√
αkρk
)i+(√
αkρk
)j
; k = l, v. ced ' g
q3@"b#% (24358) ,:%#%*;1.NU) V"#Wr Wj143@"i/)9'*@U-* 'o #%3(:%'bX o # :94+
o xk + ] ?%
L0 = max
(|Uv · η| +
∣∣∣∣√γp
ρv
∣∣∣∣ , |Ul · η| +∣∣∣∣√γp
ρv
∣∣∣∣).
#%3(*(3L)b)9j;3*U'"b./91#W)(b?#%*85j1-("*;
4)5'"b"11''bFX o # :94+
o xk + ] ' ! &K o 3_?03K)*93@j,843./9'*;243'8#%P("43-* j19.F W\Ea#%3* ' *#P('.N1D
2D
1#%./#%::''Y#%>1(#%.N#::'''+R#%>#%j,+ 203(F*)Z#% `3(73(*L*)Z9",.N#HG'J*;1.N9"203( X xy#3 :9 ] )#%AN/?,*;BP'b1AG0203('$./#HG%'('N#%A$)(#%9'-* H
uk =Ai(uk)i + Aj(uk)j
Ai + Aj
; k = l, v, ced ' d g
vk =Ai(vk)i + Aj(vk)j
Ai + Aj
; k = l, v, ced ' g#JI
Ai
Aj
#%A *a'?'./'A8'*)'V'3('Ci
Cj '
1 C DF>?G < 9?> 9VC>?; @ <CA9 6@ 5?; >?D^ (203(D&j1-:'./'A/)'/)'(9'/P-;9E;3*bBK.N9"1#0)&)&a'*;3*P,"#% ))'(9/X 0k 9 ]
ρv → ρv
ρ0v
, ρl →ρl
ρ0l
, cpd ' 9 g ?%'
ρ0v
ρ0
l
;#%Ak)35+$)(';"9k ,*""9'*;"243y*;'a'"@?%'./'A)(8B1-;8:A,i''3(c ?,a'3* g B51-;O243)
'^ 6#7ST*""#6)('8)('35+6)@"9
ε =ρ0
v
ρ0l
'(,j1-A
243)-(yV./#W) ')1-;243V'#%;)9*9,+(BU1-,8243)';y"*-'y)8-* *"a#%*.DBF15?,a'3*
ε << 1 'U5 UBN.F,*' #%P'5A) 9'*;U)5$.F((''*;
3@?,A"MHJ(W , η
)= J0
(W , η
)+ εJ1
(W , η
). cpd ' g
d
" "
Qw')'3 +6.F,*'' J0 J1
#%AV)#%(9' -,*
J0 (W, η) =
0 ηx ηy 0 0 0
−uvUv + c2vηx uvηx + Uv uvηy 0 0 0
−vvUv + c2vηy vvηx vvηy + Uv 0 0 0
0 0 0 ηx ηy 0
0 0 0 −ulUl + c2l ηx ulηx + Ul ulηy
0 0 0 −vlUl + c2l ηy vlηx vlηy + Ul
J1 (W, η) =
0 0 0 0 0 0
0 0 0 γP
ρ0vηx 0 0
0 0 0 γP
ρ0vηy 0 0
0 0 0 0 0 0
αl
αv
γP
ρ0vηx 0 0 0 0 0
αl
αv
γP
ρ0vηy 0 0 0 0 0
?%'
cv =
öP
∂ρv
=
√γP
ρv
, cl =√
Θ
Uk = ukηx + vkηy. cpd ' f g
QR.F,*' J0 (W, η);t):%#%"Pq3*
Rt'v?,'3*v*#%(*'q;#%At)#%(9''q* H
λ1 (W, η) = Uv
λ2,0 (W, η) = Uv − cv
λ3,0 (W, η) = Uv + cv
λ4,0 (W, η) = Ul
λ5,0 (W, η) = Ul − cl
λ6,0 (W, η) = Ul + cl
#%3(*/D.N#0) '$243&#%3(N ?#%/'#%;)9*9
ε';/3l-*.'"*;$"* ''b`"@ +vk'/)
*)Z#%*)(*5×10−2 a#%3*v3b:%9'9*",'3*q)(k?,`3*q
10−3 )-qk t,*r 3'QR.N,"*;'
J (W, η)'3/`*;"3(*P-,#%K9'*;7)bB$.N,* J0 (W, η)
+w)-U/'3 3)bj1'*;j1'*O5;:%5)(/$.F!"*' J (W, η)
#% `3(3(";'* 5:%5)3 '#4?'"'3(* d
" V
J0 (W, η))-8) 9jaL*9)"3*)3&j19.F (\&*)Z9j,`(*9')('3*A) 9'*@)
B7Se #%&3@?,4MH
W nij =
1
2
(W n
i +W nj
)− 1
2sgn
(J0
(W , η
)) (W n
j −W ni
) cpd ' f HgV*)Z9"aL'#%*;*'"3* A) 9'*;)UB5.NQ''*83@?,4MH
W n+1i = W n
i − ∆t
Ai
∑
j∈Ni
|eij|G(W n
i ,Wnj , ηij
)+ ∆t(S1)
ni cpd ' f g
#%3(S1)
ni =
1
∆tAi
∫
Ci
∫ tn+1
tnS1 (x, y,W (x, y, t))dtdxdy,
Cj
#48V?#%3./'V)O'#%A"* U?%#V)Ci
G(W n
i ,Wnj , ηij
) 'B5ST#%"#%)LI-35+643./9'*;243L)#%(9'-*
G(W n
i ,Wnj , ηij
)= F
(W n
ij
)ηij,x +G
(W n
ij
)ηij,y cpd ' f%f%g
d
X o xk + ] 'o #3:%'+
'xy#%3,'*'+
'8t3(G0;'+
'-*;G0)*'+
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'QRy#02 +
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' ?0;#%&\'j1*j1XW 89?%#%a'./'AYq'"*;'@"9)( -*',+ 'X o [ ']
'o '*3()# )
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