Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography

Authors:
Ulrik S. Fjordholm;Siddhartha Mishra;Eitan Tadmor
Affiliations:
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG J 48, Zürich 8092, Switzerland;Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Zürich 8092, Switzerland;Department of Mathematics, Center of Scientific Computation and Mathematical Modeling (CSCAMM), Institute for Physical Sciences and Technology (IPST), University of Maryland, MD 20742-4015, USA
Venue:
Journal of Computational Physics
Year:
2011

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Abstract

We consider the shallow water equations with non-flat bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states of lake at rest as well as moving equilibrium states. We design energy conservative finite volume schemes which preserve (i) the lake at rest steady state in both one and two space dimensions, and (ii) one-dimensional moving equilibrium states. Suitable energy stable numerical diffusion operators, based on energy and equilibrium variables, are designed to preserve these two types of steady states. Several numerical experiments illustrating the robustness of the energy preserving and energy stable well-balanced schemes are presented.