Numericalmodellingof tsunamimitigationbymangroves

Putu Harry Gunawan1,2,∗, D. Doyen1, R. Eymard1, S.R. Pudjaprasetya2LAMA (Laboratoire d’Analyse et de Mathématiques Appliquées) UPEM1 & ITB2

∗putu-harry.gunawan@univ.paris-est.fr

Problem

The role of mangroves (coastal forests) in themitigation of tsunami impacts is a debatedtopic and numerical simulation can bringan important contribution to this debate.Several articles have been devoted to thenumerical modelling of tsunami mitigation bymangroves in the past few years [1, 2, 3].The arrival of a tsunami in a coastal area isgenerally described by the shallow water model.

Shallow water equationsA way of modelling the mangrove resistanceis to add a bottom friction term (similar tothose used in the simulation of overland flows(Delestre)[4] ) to the shallow water equations:

∂th+ ∂x(hu) = 0,

∂t(hu) + ∂x(hu2 + 1/2gh2

)+ gh(∂xz + Sf ) = 0,

where h is the water height, u is the velocity,g is the gravitational constant, z is the topog-raphy of the bottom, and Sf is a friction termdue to the mangrove. A usual friction term is,for instance, the so-called Manning friction orDarcy-Weisbach:

Sf = Cfu|u|h4/3

, or Sf = Cfu|u|8gh

,

where Cf is a given coefficient.

References[1] H. Yanagisawa et all, The reduction effects of man-

grove forest on a tsunami based on field surveys atPakarang Cape, Thailand and numerical analysis, Es-tuarine, Coastal and Shelf Science 81 (27-37), 2009.

[2] H.L. Koh, et al, Simulation of Andaman 2004tsunami for assessing impact on Malaysia, Journal ofAsian Earth Sciences 36 (74-83), 2009.

[3] S.Y. Teh et al, Analytical and numerical simula-tion of tsunami mitigation by mangroves in Penang,Malaysia, Journal of Asean Earth Sciences, pp. 38-46,2009.

[4] O. Delestre, Simulation du ruissellement d’eau depluie sur des suface agricoles, PhD thesis, UniversitéD’orléans. 2010.

[5] F. Bouchut, Nonlinear Stability of Finite VolumeMethod for Hyperbolic Conservation Laws and Well-Balanced Schemes four Sources, Birkhauser Verlag,Germany. 2004.

[6] E. F. Toro, Riemann Solvers and Numerical Meth-ods for Fluid Dynamics A Practical Introduction 3rdedition, Springer-Verlag: Berlin Heidelberg, 2009.

[7] R. Herbin, et al, Consistent explicit staggeredschemes for compressible flows - Part I: the barotropicEuler equations, submitted, 2013.

[8] D. Doyen, and P.H. Gunawan, Explicit staggeredscheme for the shallow water equations with topogra-phy, submitted, 2014.

Numerical method

Generally, all the unknowns of the system are approximated on the same mesh and the numericalfluxes are computed with an approximate Riemann solver. We refer to Bouchut [5] andToro [6] for a thorough description and analysis of this approach. Staggered finite volumediscretizations for solving nonlinear hyperbolic system of conservation laws have been investigatedmore recently Herbin et all [7] and Doyen et all [8]. The numerical fluxes can then be computedsimply componentwise, using upwind or centered approximations.We consider the time interval (0, T ) and the space domain Ω := (0, L) with solid wall boundaryconditions (i.e. u = 0 at each end of the domain Ω). The time interval is divided into Nt time stepsof length ∆t and, for all n ∈ 0, . . . , Nt, tn := n∆t. The domain Ω is divided into Nx cells of length∆x. The left end, the center and the right end of the i-th cell are denoted by xi− 1

2, xi and xi+ 1

2,

respectively. We set M := 1, . . . , Nx, Eint := 1, . . . , Nx − 1, Eb := 0, Nx, and E := Eint ∪ Eb.The mass conservation equation is discretized with an explicit upwind scheme:

hn+1i − hni +

∆t

∆x

(qni+ 1

2− qni− 1

2

)= 0, ∀i ∈M,

where

qni+ 12

:= hni+ 12uni+ 1

2, hni+ 1

2:=

hni if un

i+ 12

≥ 0

hni+1 if uni+ 1

2

< 0, ∀i ∈ E .

The momentum balance equation is discretized with explicit upwind fluxes for the convection termand implicit centered fluxes for the pressure term and topography term:

hn+1i+ 1

2

un+1i+ 1

2

− hni+ 12uni+ 1

2+

∆t

∆x

[qni+1u

ni+1 − qni uni +

1

2g[(hn+1

i+1 )2 − (hn+1i )2

]+ ghn+1

i+ 12

(zi+1 − zi)]

+∆tgCf(hn+1i+ 1

2

)1/3 |uni+ 12|un+1

i+ 12

= 0, ∀i ∈ Eint,

wherehni+ 1

2:=

1

2

(hni + hni+1

), ∀i ∈ Eint,

qni :=1

2

(qni− 1

2+ qni+ 1

2

), uni :=

uni− 1

2

if qni ≥ 0

uni+ 1

2

if qni < 0, ∀i ∈M.

At each time step, the Courant number is defined by

ν :=∆t

∆xmaxi∈M

(∣∣qni+ 1

2

+ qni− 1

2

∣∣2hni

+√ghni

).

The numerical simulations show that the staggered scheme is stable under the CFL condition ν ≤ 1.

Numerical simulations