Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography

Andreas Hiltebrand,Siddhartha Mishra

doi:10.3934/nhm.2016.11.145

Abstract

We describe a shock-capturing streamline diffusion space-time discontinuous Galerkin (DG) method to discretize the shallow water equations with variable bottom topography. This method, based on the entropy variables as degrees of freedom, is shown to be energy stable as well as well-balanced with respect to the lake at rest steady state. We present numerical experiments illustrating the numerical method.

Highlights

The shallow water equations model many phenomena of interest to meteorology and oceanography
The shallow water equations with bottom topography are an example of nonlinear systems of balance laws of the form, d
We extend the shock-capturing streamline diffusion space-time discontinuous Galerkin (DG) method of [6, 7] to approximate the shallow water equations with variable bottom topography

Summary

Introduction

The shallow water equations model many phenomena of interest to meteorology and oceanography. In one space-dimension, the shallow water equations with variable bottom topography are given by, ht + (hu)x = 0, (hu)t + gh2 + hu x = −ghbx. H is the water height, u is the depth-averaged velocity, g the gravitational constant, and b is the bottom topography. The two-dimensional version of the equations is provided in the appendix. The shallow water equations with bottom topography are an example of nonlinear systems of balance laws of the form, d. Ut + Fk(U)xk = S(U) (2) k=1 by defining

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Conclusion