Numerical evidence is presented to demonstrate that state of the art numerical schemes need not converge to entropy solutions of systems of hyperbolic conservation laws in several space dimensions. Combined with recent results on the lack of stability of these solutions, we advocate the more general notion of entropy measure valued solutions as the appropriate paradigm for solutions of such multi-dimensional systems. We propose a detailed ...

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.