The Runge-Kutta local projection p1-discontinuous-Galerkin finite element method for scalar conservation laws

Abstract

In this work we introduce and analize a new explicit method for solving numerically scalar conservation laws. The time-discretization of the method is based on a second order accurate TVD Runge-Kutta technique (used recently by Osher and Shu to solve scalar conservation laws in the framework of finite difference schemes), while the space-discretization is based on a discontinuous finite element method for which the approximate solution is taken to be piecewise linear in space (i. e., the local projection P0P1 -discontinuous Galerkin method introduced recently by Chavent and Cockbum). The resulting scheme is TVBM (total variation bounded in the means), converges to a weak solution, and is formally second order accurate in time and space for cfl ε [0, 1/3]. We give extensive numerical evidence that the scheme does converge to the entropy solution, and that the order of convergence away from singularities is optimal; i. e., equal to 2 in the norm of L∞(L∞loc).