Discontinuous Galerkin methods for first-order hyperbolic problems

Abstract

In this paper we consider discontinuous Galerkin (DG) finite element approximations of a model scalar linear hyperbolic equation. We show that in order to ensure continuous stabilization of the method it suffices to add a jump-penalty-term to the discretized equation. In particular, the method does not require upwinding in the usual sense. For a specific value of the penalty parameter we recover the classical discontinuous Galerkin method with upwind numerical flux function. More generally, using discontinuous piecewise polynomials of degree , the familiar optimal error estimate is proved for any value of the penalty parameter. As precisely the same jump -term is used for the purposes of stabilizing DG approximations of advection-diffusion operators, the discretization proposed here can simplify the construction of discontinuous Galerkin finite element approximations of advection-diffusion problems. Moreover, the use of the jump-stabilization makes the analysis simpler and more elegant.