Energy preserving and energy stable numerical schemes for the shallow water equations

S. Mishra, CMA, Oslo, Norway

Monday, December 8
at 09.15, HG E 1.2

Many systems of conservation laws arising in Physics are equipped with an entropy function and entropy fluxes. The entropy framework singles out physically admissible solutions and provides stability estimates particularly for multi-dimensional systems. Standard numerical schemes of the finite volume type are not necessarily entropy stable, particularly at higher-order of accuracy.

We describe a new framework for designing finite volume schemes that are entropy stable. The basis of this framework is the design of entropy conservative schemes for systems of conservation laws. These schemes are combined with either explicit physical diffusion operators or novel numerical diffusion operators based on entropy variables. The resulting schemes are entropy stable.

We illustrate this framework in the case of shallow water equations in multi-dimensions. In this case, the total energy serves as the entropy function. We design different energy preserving schemes and couple them with numerical diffusion operators to obtain energy stable schemes. Similar schemes are designed for shallow water equations with topography. The resulting schemes preserve interesting steady states. The performance of these schemes are illustrated by several numerical experiments. The talk is based work jointly done with E. Tadmor (Univ. Maryland, S.A) and U . S. Fjordholm (Univeristy of Oslo, Norway)