A new theoretically motivated higher order upwind scheme on unstructured grids of simplices

Abstract

Many approaches exist to define a cell-centered upwind finite volume scheme of higher order on an unstructured grid of simplices. However, real theoretical motivation in the form of a convergence result does not exist for these approaches. Furthermore, some theoretical results of convergence exist for higher order finite volume methods, where no description of the numerical implementation is given to realize the necessary requirements for the convergence theory. Therefore we present in this paper a new limiter function which is motivated by these requirements and ensures a convergent scheme in the theoretical context: The approximated solution converges to the entropy solution in the case of scalar conservation laws in two space dimensions. This new limiter function is combined with a typical class of reconstruction functions very efficiently, which is illustrated by several test examples for scalar conservation laws as well as systems of such laws. In connection with the requirements to be fulfilled, a proof of a maximum principle of the finite volume scheme applied to simplices and dual cells is given. So for the approach of the higher order upwind finite volume scheme on dual cells, as used in several papers, a missing proof is now given. The ideas in this proof are also applied to the discontinuous Galerkin method, so that an existing maximum principle can be improved considerably. The main advantage comes from the fact that no requirements on the discretization of the domain are necessary: no B-triangulations or Delaunay triangulation are needed.