Tsunami Modelling using the Well-Balanced Scheme

A well balanced numerical scheme based on stationary waves for shallow water flows with arbitrary topography has been introduced by Thanh et al. [18]. The scheme was constructed so that it maintains equilibrium states and tests indicate that it is stable and fast. Applying the well-balanced scheme for the one-dimensional shallow water equations, we study the early shock waves propagation towards the Phuket coast in Southern Thailand during a hypothetical tsunami. The initial tsunami wave is generated in the deep ocean with the strength that of Indonesian tsunami of 2004.

Authors:

• Ahmad Izani M. Ismail
• Md. Fazlul Karim
• Mai Duc Thanh

Keywords:

• Tsunami study
• shallow water
• conservation law
• well-balanced scheme
• topography. Subject classification: 86 A 05
• 86 A 17.

References:

[1] N. Andrianov and G. Warnecke, On the solution to the Riemann problem
for the compressible duct Flow, SIAM J. Appl. Math., 64(3):878-901,
2004.
[2] N. Andrianov and G. Warnecke, The Riemann problem for the Baer-
Nunziato two-phase flow model, J. Comput. Phys., 195(3):434-464,
2004.
[3] E. Audusse, F. Bouchut, M-O. Bristeau, R. Klein, and B. Perthame, A
fast and stable well-balanced scheme with hydrostatic reconstruction
for shallow water flows, SIAM J. Sci. Comp., 25(6):2050--2065, 2004.
[4] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic
conservation laws and well-balanced schemes for sources, Frontiers in
Mathematics. Birkhauser Verlag, Basel, 2004.
[5] R. Botchorishvili, B. Perthame, and A. Vasseur, Equilibrium schemes for
scalar conservation laws with stiff sources, Math. Comput., 72:131--
157, 2003.
[6] R. Botchorishvili and O. Pironneau, Finite volume schemes with
equilibrium type discretization of source terms for scalar conservation
laws, J. Comput. Phys., 187:391--427, 2003.
[7] P. Goatin and P.G. LeFloch, The Riemann problem for a class of
resonant nonlinear systems of balance laws, Ann. Inst. H. Poincaré
Anal. NonLineaire, 21:881--902, 2004.
[8] L.Gosse, A well-balanced flux-vector splitting scheme designed for
hyperbolic systems of conservation laws with source terms, Comp. Math.
Appl., 39:135--159, 2000.
[9] J.M. Greenberg and A.Y. Leroux, A well-balanced scheme for the
numerical processing of source terms in hyperbolic equations, SIAM
J. Numer. Anal., 33:1--16, 1996.
[10] J.M. Greenberg, A.Y. Leroux, R. Baraille, and A. Noussair, Analysis
and approximation of conservation laws with source terms, SIAM J.
Numer. Anal., 34:1980--2007, 1997.
[11] E. Isaacson and B. Temple, Nonlinear resonance in systems of
conservation laws, SIAM J. Appl. Math., 52:1260--1278, 1992.
[12] E. Isaacson and B. Temple, Convergence of the 2x2 Godunov method for
a general resonant nonlinear balance law, SIAM J. Appl. Math.,
55:625--640, 1995.
[13] S. Jin and X. Wen, An efficient method for computing hyperbolic
systems with geometrical source terms having concentrations, J.
Comput. Math., 22 (2004), pp.~230--249.
[14] D. Kröner and M.D. Thanh, Numerical solutions to compressible flows
in a nozzle with variable cross-section, SIAM J. Numer. Anal.,
43(2):796--824, 2005.
[15] P.G. LeFloch and M.D. Thanh, The Riemann problem for fluid flows in a
nozzle with discontinuous cross-section, Comm. Math. Sci., 1(4):763--
797, 2003.
[16] P.G. LeFloch and M.D. Thanh, The Riemann problem for shallow water
equations with discontinuous topography, Comm. Math. Sci., (to
appear).
[17] D. Marchesin and P.J. Paes-Leme, A Riemann problem in gas dynamics
with bifurcation. Hyperbolic partial differential equations III, Comput.
Math. Appl. (Part A), 12:433--455, 1986.
[18] M.D. Thanh, M. F. Karim., and A. Izani Md. I., Well-balanced scheme
for shallow water equations with arbitrary topography, Inter. J. Dyn.
Sys. & Diff. Eqs. (accepted for publication).