A Discontinuous Finite Element Method For Scalar Nonlinear Conservation Laws

Abstract

Solutions of scalar nonlinear conservation laws are calculated by using discontinuous finite elements in one or in several dimensions. The standard first order finite difference scheme is obtained with piecewise constant approximations, while higher degree piecewise polynomial approximations give more accurate schemes. At discontinuities of the approximate solution, numerical fluxes are calculated by one-dimensional approximate Riemann solvers. The method is stabilized with truely multidimensional slope limiters. Special attention is given to piecewise linear approximation which is shown to be total variation diminishing and con-vergent. In two dimensions numerical experiments are presented on structural as well as on unstructured meshes.