In this paper, we present numerical methods for the dam break problem for shallow water equations. The particular method we considered here is the adaptive moving-grid finite volume approach, based on the equidistribution of compute nodes using a control (or monitor) function. It is one of the moving-grid methods proposed by Huang et al. [22]. Since the computational accuracy of such methods strongly depends on the choice of the control function, we propose a well-suited control function allowing to obtain a better convergence rate. Motivated by efficiency considerations for problems in one or more spatial dimensions, we will apply the proposed numerical approach for the solution of the dam break problem for the shallow water equations, widely used for modeling the flow of water rivers, lakes, reservoirs, coastal areas and other situations in which the water depth is much smaller than the horizontal length scale of the flow. The performance of this framework is illustrated for numerical experiments that support the main theoretical results.

shallow water equations, dam break problem, grid technique, finite volume method.

Received: December 21, 2022; Accepted: January 27, 2023; Published: February 7, 2023

How to cite this article: Diakalia Koné, Abdoulaye Samaké, Yaya Koné and Lamien Kassienou, A numerical framework for the dam break problem for shallow water equations using moving-grid finite volume method, International Journal of Numerical Methods and Applications 23(1) (2023), 87-106. http://dx.doi.org/10.17654/0975045223005

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] H. Tang and T. Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41(2) (2003), 487-515.
[2] W. Huang and R. D. Russell, Adaptive Moving Mesh Methods, Vol. 174, Springer Science and Business Media, 2010.
[3] E. Audusse, Modélisation hyperbolique et analyse numérique pour les écoulements en eaux peu profondes, Ph.D. thesis, Paris, 6, 2004.
[4] K. Korichi, A. Hazzab and A. Ghenaim, Schémas à captures de chocs pour la simulation numérique des écoulements à surface libre, LARHYSS Journal 8(1) (2010), 81-100.
[5] E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Ellipses, 1991.
[6] M. Ohlberger, Adaptative finite volume methods for displacement problems in porous media, Comput. Visuel Sci. 5(2) (2002), 95-106.
[7] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-balanced Schemes for Sources, Springer Science and Business Media, 2004.
[8] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Vol. 31, 2002.
[9] P. Lagrée, Résolution numérique des équations de saint-venant, mise en oeuvre en volumes fnis par un solveur de riemann bien balancé, Institut Jean Le Rond D’Alembert, 2020.
[10] W. Dai and P. R. Woodward, A simple Riemann solver and high-order Godunov schemes for hyperbolic systems of conservation laws, J. Comput. Phys. 121(1) (1995), 51-65.
[11] R. Eymard, T. Gallouït, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math. 92(1) (2002), 41-82.
[12] S. Godounov, Résolution numérique des problemes multidimensionnels de la dynamique des gaz, Mided, 1979.
[13] E. Audusse and M.-O. Bristeau, Transport of pollutant in shallow water a two time steps kinetic method, ESAIM: Mathematical Modelling and Numerical Analysis 37(2) (2003), 389-416.
[14] A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23(8) (1994), 1049-1071.
[15] L. Kassiénou, S. Longin and O. Mamadou, Using the adaptive mesh finite volume method to solve three test problems, Far East Journal of Applied Mathematics 95(4) (2016), 283-310.
[16] V. D. Liseikin, Grid Generation Methods, Vol. 1, Springer, 1999.
[17] J. M. Stockie, J. A. Mackenzie and R. D. Russell, A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 22(5) (2001), 1791-1813.
[18] W. Huang and R. D. Russell, Adaptive mesh movement - the MMPDE approach and its applications, J. Comput. Appl. Math. 128(1-2) (2001), 383-398.
[19] A. van Dam and P. A. Zegeling, A robust moving mesh finite volume method applied to 1d hyperbolic conservation laws from magnetohydrodynamics, J. Comput. Phys. 216(2) (2006), 526-546.
[20] W. Huang, Practical aspects of formulation and solution of moving mesh partial differential equations, J. Comput. Phys. 171(2) (2001), 753-775.
[21] D. J. Higham and N. J. Higham, MATLAB Guide, SIAM, 2016.
[22] W. Huang, Y. Ren and R. D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM J. Numer. Anal. 31(3) (1994), 709-730.