Un schéma aux volumes finis avec matrice signe pour les … · 2017. 1. 17. · Un sch ema aux...

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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m

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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)62IADCJCK4Â¬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$ÐL@|Ì ÊAEM)EÑÏ+=XMA?AFA=^Å Ïª=XMhORz¿XÏ+=M =hE~D E~FÐ?)D6GJIHGJLUFzD ?AFAE<A@6E~ÐGHÍ$@|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$ÌªIJEÌ 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

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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= _¿@[email protected] 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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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 ÏªÊAGHM&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$F)E$ ÏªÀ 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 `ÏªÀ 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$ÌªD=?)DEªÌ$=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?Ð´L1ÇGH?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_DÑÏª=XMAIJE 6ÉzGRQR `ÏªÀ 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)) ,

. >

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

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1AnalytiqueSRNHSHybride

! #"$%&'( *),+.- '0/214365 "

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

78!9 ;:=<?>4 '(@'( *)AA>4 B& CED ' FHGI*) ' J%

αn

i+ 1

2

- 'I/213%5"

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

K6LM N:O4 B '$(6'$( )QA> R&S BCED ' $

θni+ 1

2

- '0/213%5 "

LUT

D D

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1AnalytiqueSRNHSHybride

º?> ! #"%$ &

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

78'9 ;:=<?>4 '(@'( *)AA>4 B& CED ' FHGI*) ' J%

αn

i+ 1

2

- 'I/213 ("

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

K*)M N:O4 B '$(6'$( )QA> R&S BCED ' $

θni+ 1

2

- '0/213 ( "

−11 −10 −9 −8 −7 −6 −5 −4−10

−9

−8

−7

−6

−5

−4

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@

: 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=FAÌª$z?AGHÅULIHE$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


(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


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


8 ( ) ' "

'

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$@É_D-IJE D@|=CGJ|GJÍ$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$MAEÌ ÊA=ÌU¿<AL@Ì~=CFD@|EÍJEÓMAE~?ÇGJÍ~ÐEÌ Ê&L<pÌ$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

=D6=CFAz?AEÍJE6Ì ÊA~´L # » »Á <a=C?A@Ì$E´Ì L~¿y E$DL ÅC$@| FA=CF E~FD@6=<AGJz?AEÐ;~ÐE´L<A@6Í~Ì$=C@|@6E~Ì~DGH=CFE$FD@|=C<AGHX?)E E$KC º b,X7;X7v] 2&/^];X/1#)"$*J7Ö\_#)"_\^/1b& 5$#h7ËÖLFAfÌ$EªÌ$LGJIdÈLvMAE~?Ç Ì ÊA=XÌ$z?AE6Eª<A@6=<&LKCE~FzDfMAEªIJLKL?AÌ ÊAEªÅCE~@6fIÃLMA@|=CGHD6E º

~DLUDKL?AÌ Ê)E x ≤ 5 _D LUDfMA@|=CGHD x > 5ρl = 1 ρr = 1ul = 300 ur = −300pl = 105 pr = 105

>

DDI D 1D

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1SRNHSRusanov

8 ?)4" (@' < 4 ( G /2143 '0/2143 1*15 5 111> ( ) ' "

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10x 10

4

SRNHSRusanov

M L 8 >P"" ( ) 4 ( G /213 '0/213 1145 5P1*11> ( ) ' "

0 1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

5

M N>P"" ( *) 4 ( G /213 M'0/213 11 5 111> ( ) ' "

0 1 2 3 4 5 6 7 8 9 10−300

−200

−100

0

100

200

300

"!$#&%'#&(*),+- /.10" 2"+3.1+4!$#5), 76 8%9),+$:;<6>=@?A B:#C=@?A ?$?$D8:FE?"?"?HGI0"%J!$#LK

M

: D D

O=

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QP

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ÅC=I º . > ¿&ÆIJIJGH<A6E~$¿,yL@|GJ$¿¬+RQR) º`ÏªÀ RQ= ¿ #H ´7_! 0]6*_''a/KJC 0 0/1bR/+2 '&7_~"_/1*. 0]+\m\_ 7 V\Ó/Ó] /1b)\^7_=1 0/1b

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ÎÎ @ . ¾ ºÐÎ LU@|DE~F±LF)M )ºÐ¹Rº´Î ÈX´LFd¿A 7_* ="!#\_ cb ! 7 ´7_c2&/'1\ /1 /1bd7" 0 ´7_b)\_ 0/1ba*¬2!'&7_~"_/1*. 0]´] /1b)\^7_=1 0/1bR*,!y\ ¿ )º¬Á =C<A?)D º ,ÊzÈX º $ ! RQ@. _¿FA= º X¿h.@O=QR º

ÎÎ ÔPO= ¾ ºyÎ L@DE~F³¿ )ºy¹RºyÎ ÈX´LUF³¿yLUFAM , º Ë º ÔLmÇd¿ bAb& c 7"/ Ö7_7_bd] 7´_''a/KJC 0" 0/1b)\Ñb,´7_b&/ ',] /1b, c 0/1b)\ /1(\ 2&/^] ^\ ¿ Á =C º ,?A@|E¾<A<AI ºX¹ LUDÊ º A +R(PO= !¿AFA= º .¿aOR([email protected]¿ G,DÊ LF L<A<aE$F)MAG,ÇBÈÄ º ÒªE_È7AD º

Ô = , º ÔZL1ÇLFAMÄ º RE$FAMA@|= ¿ m\_ 7 V\p/ ] /1b)\^7_1 0/1b *,!y\ ¿ Á =C º ,?A@|E¾Ñ<)<AI ºa¹ L1DÊ º % RQ=C _¿&XP@).QP º

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XE$K º E~KC?AGHF³¿B /'X%_*. J\$ 0/1b 7_\_ 0 v#X*, 0/1bb&# ´%__ 5$#&7 X7 \ª% ] /1#*J7 ´7_b&c\ `'&2)"\_ 5$#&7 \ ¿ ,Ê º Ë º D6ÊAE$|GJ~¿ FAG,ÅCE$@|6G,DªMAE,@6=^ÅCE~FAÌ$EU¿³ º

&=C@VR Æ º)» º &=C@|=A¿ 07 b&bV'a/"$*J7 V\+b,c2&7 ´7_c2&/' /1\^/1*6^ cbc2&7Ö7!®/" 0 ´7_b)\_ 0/1ba* \ 2)*c*J/! ! 7_ 765$#) 0/1b)\ ¿ ,ÊAGJIH=C º &@6LFA º À=^È º =XÌ º Ô¬=CFAM)=CFE~@ º ¾ RQRC _¿FA= º = > R)¿ > .@ =Q@ º

&=C@VRQR ÆIJE~?)DE~@6GJ= » º &=@6=A¿ 07 b&b\^/1*67_\b,Ób&#H ´7_! 0]6*< ´7_c2&/'1\ /1 # 7 "ba v 0] \ ¿&6E~Ì$=CF)M;E$M º ¿ <A@6GHFAKCE~@ (#E~@6IJLKA¿&Ä®E~@6IHGJF³¿¬RORQR)¿&¾ <A@LÌ_DGHÌ LI¬GHFzD6@6=XMA?AÌ DGH=CF º

O@

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W

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!8+O4 !'/:52:S_0/&RS!^:]! 3 "M' A 51706&SV?XW;:U=`9"^.:51^ "!31mh~4-qp0º '/e.3&4:40:U.a "!LC9 "L51 "U.34 RS!(! '-352.4:5bq./RS!0(!T9": "B52 -04|0&!13452_4:.4TY51:30(*MY "!&/$g !B !&9"!234526&<=HJI 6aº~B !&9"!3452L6&<=HJIJ6c! @ 0B3357M&'N:b`9b'_345204:U.34F51 "U]!9`!0:]RS!]!&.'/4:1E%52.3/:702<= -P52/:> @ P04:U.351UZ51 "U`P- "'!&. "b0 <=52F -4:CK\P85b/~0&[ A Z0~'ND52 A 0(/U.3 80!&/^')( & `+8/52. "0f'NB "'!&. "e0&!3 "M' A 52^0D<=52e`9"f4::0_.:51]3 "!3ºx "!&~4:9'!i'G(651`4Z0& \.Z524\. -0Z "!~'/' "; "5151:~')(*4\.!0&J0!F34526&<=HJI 607'b`7'N10>'/e351 A b3\./ ³º$3!F "!7'/' "]'G(*'?RS!;Y "!'T "M&' A 51~0&J68/U3?GW;S$C9!T "Q @ .&/i/34 @ !&'/ A UºH= "!;''/ "52 S:'/i:Y:0|'N ?3 "4:4i0~'NZ: "339/ ">0&!7`3q./ /U.3 80!&.Jh~357j0:k7:ZW_lk:R-!&kmhiW_noPrh~ |W_n"pºrb3/57!&'N/ "S!5143/RS!70fRS!'R-!&`i3.RS!f4\.4Z52:4=2:[Q:!UT')(*,+83/ "10!F3452768<ZHZI 62g]'G( "0&0!-+31!&/'/.UJ'Nb.:R-!&7"98J6|$Pv:]'NFRS!'/`/ "L0&! 345268<ZHZI 6V=#%._T'/ .:4]0&= "!3MQ:0(*:3!3$º

:<;>= ?A@ y @CB6@^z\uED swDs Fw wCGIHsJD

. e: "0 A 3^'/_3 "M' A 52J "'//&4` `9^.352^3 "!:^!9SCK

∂u

∂t+∂f(u)

∂x= s(x, u),

u(x, 0) = u0(x)

4Lu : R × R

∗ −→ RP(x, t) −→ u(x, t)

Pf : R −→ R

0&b'/3 C1 :s(x, u)

\D'.:51]3 "!3º

"

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v$4:5b09" '/!51&/6&<=HJI M4;!&v'G(*!&. /51/'/.4v0$'/| "'!&. " 0!^3 "M' A 510]<=52 e 52 @A P(4:=0^'/75b& A 3_!9U.!K

uni+ 1

2

= 12

(un

i+1 + uni

)−

αn

i+12

2Sn

i+12

(f(un

i+1) − f(uni ))

+αn

i+12

2Sn

i+12

∆xsn(xi, xi+1, uni , u

ni+1),

un+1i = un

i − r(f(un

i+ 1

2

) − f(uni− 1

2

))

+ ∆tsni .

$1Ui "51&T0'G(*'83|#%T0;'T0!8+8 A 51=Y "!;'/T; "51 @"A & 4L

f(uni+1) 6= f(un

i )!L9'/:!b0&! .5 A .e0L: "U.'/

αni+ 1

2

=Sn

i+ 1

2∣∣∣f ′(ani+ 1

2

)∣∣∣V

4:.M'/º$:52'/ ½ S αni+ 1

2

>9'!T0'G(4:.Q_&40&/:! "eJK

uni+ 1

2

=1

2

(un

i+1 + uni

)−1

2sgn

(f ′(an

i+ 1

2

)) (un

i+1 − uni

)+

∆x

2|f ′(ani+ 1

2

)| sn(xi, xi+1, u

ni , u

ni+1)

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

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&

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∂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

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ºC &'/| @ !3JE )ºoSK(-

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2

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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

4

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

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! $%!&'(*),+ -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/$9":57<;7-.-=*?>[email protected]:C#(EDF":G/@H*JIK*.-/$

(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

"

%E * %% 2 *9**2 * *

! .,),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

%E * %% 2 *9**2 * *

:<; : 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

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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

).

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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 #"%$&)(+*0DS*~-=8.a+B/Tb;Q#B/@/" *LDt; C− [email protected]@v":B/$eBL-/"DS*L-=8.a+B/Tb;U;/PxPDF"9&)(+BOU(E2*L-=57DF(1$9":57 -/$:;#$":57+;#"%@H*4@H*/[email protected](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 [email protected]@/":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

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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=

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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

'"

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0.2

0.4

0.6

0.8

1

1.2

1.4le fondla surface libre

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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 * *

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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 #%% %

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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

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!`!&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

ul = 0m/s ur = 0m/s`9"^#d "00U/S! Ed @ !&Q1YK~:=`9":]#d "&0e'/]E @ !3 QZYCUKvY ;:Z$ "1523R-!&TR-!&Z'|.352T3 "!3.'U.$'NJ9-.3=0 "U9 @ :P-Q "!&'/7 52 @A 7'/19-3>0>: "U9" @ &BV^0>'G( "0&D0& 0.8PQ.0JRS!>Y "! '

` e 52 @A _'/D98.:]0^ S9 @ :D:3=0^'G( "0&_0 0.65Y

$34:f0^'N>52^')(* "30_0&f'/>9-.3]0^ "U9 @ :70!4:5b26&<=HJI 60/51/S!4Z::r(\+8'/RS!_T'/D0/3 "U/S!4 0_: "U\T0 -Q^'/4/3J!&9"!0'NO5b[Y . 523R-!&e!3=R-!&'NL9-30e: "U9": @ :L:3204: "U.

'

%E * %% 2 *9**2 * *

#d "\./ 0B 0/ "&7/./'/:f0! 3 "M' A 51#YL9-3c0c "U9": @ :4:.U0(*!&S='!TY:..^RS!^'/_3 "M' A 51_3='!T./0[Y k l_E= k _ k7kE kEnd 9 9Al k gx 1n s 9 kEn kkEEn#

'n

E*9 #%% %

0 2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

4

4.5

5AnalytiqueSRNHS

O ETQ! ^ $'?]< 'J1:M" km)D > ! n'

−4.5 −4 −3.5 −3 −2.5 −2 −1.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Log(dx)

Lo

g(L

1−

Err

eu

r)

pente de la vitesse = 0.835, pente de la hauteur = 0.797

vitessehauteur

qZY`o !#" ' 0.8

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18 20

AnalytiqueSRNHS

ZY$ % &'()*+,&-,./"1032,4)5!6 7 -+!#"-!#

−4 −3.5 −3 −2.5 −2 −1.5−2

−1.5

−1

−0.5

0

0.5

Log(dx)

Lo

g(L

1−

Err

eu

r)

pente = 0.6596

vitessehauteur

qZY`7 % 8 91:0/;<>=@?9 !#" ' 0.65

"

%E * %% 2 *9**2 * *

k7 # 9 l_ 46k n6 % k7 % k # Ax k # !#"$!% &!% !#"$!')(%DE*!%!

x0 = 0K

hl = 5m hr = 1m,

ul = −4m/s ur = 9m/s. +,

1 − <-)./*)"$!0 123!% 4"56788!0"$!R6PYR6!898: @ fg"$;2(=<>.?"@75"$ !#"$!0 &8!0"$!R65(% (%5 3 A (06!#2) 4"

Ul−→:?<8./*?"$!%

U2−→SSTU3−→:?<-)./*)"$!0

Um−→-?<-)./*)"$!0

Ur.

jOkgAk nk7 !#"$!% &!% !#"$!')(%DE*!%!

x0 = 0K

hl = 4m hr = 0.1m,

ul = 0m/s ur = 0m/s.B-(%6C"!% DEFC"M( A 8"G C8"$!#"$6.F"$8!%H."$)"$H 8"$ 4"$ U1

PU2

"U3

m3$Jh|&\p C ?"$.-:I- 86!02) 4"(%5 3 A (056!#2) "

Ul−→<8./*?"$!%

U1−→SSTU2−→<.?/*"!%

U3−→FJ3

Ur.

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&

E*9 #%% %

−10 −8 −6 −4 −2 0 2 4 6 8 101

1.5

2

2.5

3

3.5

4

4.5

5SRNHS (avec MUSCL)SRNHS (sans MUSCL)analytique

4 , ="<k -J &(G 'J143<5 V+<-0/\1:5&+ K33 > (&)8'?"

−10 −8 −6 −4 −2 0 2 4 6 8 10−4

−2

0

2

4

6

8

10SRNHS (avec MUSCL)SRNHS (sans MUSCL)analytique

Il , Ti8(G';"" Q'01435+-0/21:K5G+K33 > K(G)8'P"

−10 −8 −6 −4 −2 0 2 4 6 8 10−20

−15

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0

5

10SRNHS (avec MUSCL)SRNHS (sans MUSCL)Analytique

h$ L.<_]<(G' 'J143<5 V+-J/21:5&+3K3 > (&)8'?"

\A

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1.5

2

2.5

3

3.5

4

4.5

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4&A , ="<k -J &(G 'J143<5 T"M+<-0/\143<5 7+<"? %)"ga

−10 −8 −6 −4 −2 0 2 4 6 8 10−20

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5

10

x

le d

eb

it

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Il!Al L4_](Y' h'J143<5 "]+-0/21435 7<+" )"4a

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0

1

2

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4

5

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le n

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bre

de

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ud

e

SRNHSVazquezAnalytique

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&A CK<?D \[ ]?M<"<S! ^ %'?]<;+ > M)8'mK??@' 0.76

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1.5

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3

3.5


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−10 −8 −6 −4 −2 0 2 4 6 8 100

1

2

3

4

5


R C "IV MI "I %Y' (Z) .1035 7 2<+-,/.1035*4*4:9 ; L=3?

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1

2

3

4

5

6 SRNHS (avec MUSCL)SRNHS (sans MUSCL)Analytique

E R&C ( !Q"$Q%Y' (Z): .1035 7 2I+-,/.1035234*4:9<; =>?

−5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5SRNHS(sans MUSCL)SRNHS(avec MUSCL

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0

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(p− pik)∆αk

_k = l, v

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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([email protected]+RO ^6C,('!096(%)"$!0 S 36C.!%W66c(%d5DP 8 Y

0'0

7 ! (

1D

(*) 143*6 ! 6,3

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∂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&#0B1# ()"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",# -\&#1 (); /.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# &%&#1I &'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:;&#1&")(,: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&#1&G98;&r")#9# J(); \&#1 ();# &-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; %.\ &#1()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&&#198.# \ ")(,g() &'FA H\ ;N.# r HI:;:;&#1()* "324# ()&/ # r")9';9q+N "324# ;:",:&#1(,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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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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)((

kk

w

! #"$ PA# MA98.#1(,&M;;/!# .# "*GF#18:F#1(,g V98:F#% !># jB;S u98:;<#A=>?@ .#1")[0#1SJ(, ()'()"WNC&R<# (,")")# \ O5 _&;&: N1I &S ;& J")];F;!0#1"$# (,&!+ H G_]H E") () ;'&'B1#1()%?D#1JIO()'DnU")#B;&(,h1P 98:F#

(SRNHS)Ns$#E()s;&'BG();[ B# <F#1&'();(,\!?r

&:;;Y#1")&' K&:;"69#1K:&'()g;Rk&f");f&J")]f v2'Y");&f;=# ()s_5 X# sJ() ()'()")M8\]

%'& (kc*)+-,. "/ )#0',Dc

& # (,&"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)

'L"$#!'& ;JI;\< .#

RN−1 N ∂p ;'u")# &'s(,]&F p N h ;'u",F ($# ];&E V")#*F# ()",")

p

)(

u Mr

;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(_"$#DBG&# ()!'")()<8O'()g.# &'F(I");",(, # ()J", f#<&'()];&T :'&#1() v24 tO()';E;! v2w(,(6:T %"$#<'")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"$# :'&#1() 24# (,U&T"$#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

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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.# &&#1I &'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.+ ;'& &(,:;:&#1 (,") # (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&#1& 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


)un

j .

7

τnij =

∆tn

2Ai

|eij|(Sn


) 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))

&#8# :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

&#7 # 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 " # :&#8!# :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 # ## & 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

" :&#8# :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 + - 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

.


!aGV(U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(

Ai


&)|+*+- 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

∣∣∣∣

.


!aGV(U7|+*+-6>@c+3S*v( |∂Ci|(+3Go D)-6*v(

Ai


&)|+*+- 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 ( *),+$- ++.

L # # 3 # #

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√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¦

<"| 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

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)


! #"%$'&(*)()+, -.0/21435&7689':;)=<>-?@AB$!) , ?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)


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


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


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


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


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


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)


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)


-@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 =

√∂P

∂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)*'+

''[L1)-,:%(+

o'L3.7P-,*#+=[

'QRy#02 +

'; p+q)

' ;j1.N +! #"%$'&("*)+$', .-0/213$4"658709;:<$'/213 >=#?<@%A=B/. * #CD?E-F #&0/G13$H?')2:JIKCL@%/M)+ #N .-0/O,P #9RQ31S@#?*$UTE #,V?+ 'j1 ' *'a#%* + 9'-*".N4 *",;ST*;" 1('*./243L o 9*#0)(G0-./243

' ?0;#%&\'j1*j1XW 89?%#%a'./'AYq'"*;'@"9)( -*',+ 'X o [ ']

'o '*3()# )

'[U*;'B r 8 ?,*;*#+X 1K)LZ#1&$?* #C[IK/O)+ #\ZK >=%IK3)2"*9]/M^Q3$

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'o (lkm c g + hon 'X o # n,]

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Un schéma aux volumes finis avec matrice signe pour les … · 2017. 1. 17. · Un sch ema aux...

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