| Author | J. E. Flaherty, L. Krivodonova, J.-F. Remacle, M. S. Shephard |
|---|---|
| Title | Aspects of Discontinuous Galerkin Methods for Hyperbolic Conservation Laws |
| Year | 2001 |
| Journal | Finite Elements in Analysis and Design |
| Volume | - - |
| Pages | - - |
| Abstract | 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 polynominals is proportional to a linear combination of orthogonal polynominals on each element of degree p and p +1. These are Radau polynominals in one dimension. The discretization errors have a stronger superconvergence of order O(h exp 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 spacial 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 adaptative computations of Mach reflection and mixing-instability problems. |