On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface

Abstract

Well balanced finite volume methods used to solve the shallow water wave equations are designed to preserve the steady state of a `lake at rest'. Unfortunately, for problems involving wet/dry interfaces, this steady state is not preserved unless the involved quantities are reconstructed with care. We test four reconstruction options: stage and momentum (where bed is fixed); stage and velocity (where bed is fixed); stage, water height, and velocity; and stage, bed, and velocity (with modification at wet/dry interfaces). Reconstructions based on stage, water height, and velocity are shown to preserve the steady state problem and accurately solve a representative unsteady state problem, whereas the other options lead to problems in both maintaining the steady solution and accurately solving non-steady problems. Our results indicate the appropriate choice of reconstruction variables for various situations when solving the shallow water wave equations using finite volume methods. References 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 Journal on Scientific Computing , 25(6):2050--2065, 2004. doi:10.1137/S1064827503431090 A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM Journal on Scientific Computing , 23(3):707--740, 2001. doi:10.1137/S1064827500373413 S. Noelle, N. Pankratz, G. Puppo, and J R. Natvig. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. Journal of Computational Physics , 213(2):474--499, 2006. doi:10.1016/j.jcp.2005.08.019 S. Roberts, O. Nielsen, D. Gray, and J. Sexton. ANUGA User Manual . Geoscience Australia, 2009. http://datamining.anu.edu.au/anuga W C. Thacker. Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics , 107:499--508, 1981. doi:10.1017/S0022112081001882