Aspects of discontinuous Galerkin methods for hyperbolic conservation laws

We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, flux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree p piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees p and p + 1. These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order O(h(2p+1)), where h is a mesh-spacing parameter, at the outflow boundary of each element. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth. We present the results of applying the discontinuous Galerkin method to unsteady, two-dimensional, compressible, inviscid flow problems. These include adaptive computations of Mach reflection and mixing-instability problems. (C) 2002 Elsevier Science B.V. All rights reserved.