Aspects of discontinuous Galerkin methods for hyperbolic conservation laws.

*(English)*Zbl 0996.65106Summary: 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.

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 |