Aspects of discontinuous Galerkin methods for hyperbolic conservation laws. (English) Zbl 0996.65106

Flaherty, Joseph E.; Krivodonova, Lilia; Remacle, Jean-Francois; Shephard, Mark S.

Finite Elem. Anal. Des. 38, No. 10, 889-908 (2002).

Summary: 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.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35L65	Hyperbolic conservation laws
76M10	Finite element methods applied to problems in fluid mechanics
76N15	Gas dynamics, general

Keywords:

discontinuous Galerkin method; hyperbolic systems of conservation laws; error estimation; superconvergence; compressible inviscid flow problems; Mach reflection; mixing-instability problems