high-order adaptive and paralleldiscontinuous galerkin m：高阶自适应paralleldiscontinuous galerkin m

High-Order Adaptive and ParallelDiscontinuous Galerkin Methodsfor Hyperbolic Conservation LawsJ. E. Flaherty, L. Krivodonova,J. F. Remacle, and M. S. ShephardScientific Computation Research CenterDiscontinuous Galerkin Method Arbitrary order: extends finite volume method Structured or unstructured meshes No need for inter-element continuity Simplifies adaptive h- and p-refinementDiscontinuous Galerkin Method Face-based communication Simplifies parallel computation Sharp capturing of discontinuities Element level conservation A posteriori error estimatesDiscontinuous Galerkin Method Face-based communication Simplifies parallel computation Sharp capturing of discontinuities Element level conservation A posteriori error estimatesHowever: More mesh unknowns than FEM for same order Possibly OK with parallel computation Monotonicty control (limiting) is difficultDG Formulationl Conservation lawl Construct a Galerkin problem on j cf. Cockburn and Shu (1989) DG SolutionlSolving the Galerkin problemIntegral evaluationTime integrationFlux evaluation, limitingApproximationHigher order equationsl Discontinuous approximations needs regularization for gradientsExampleApproximationl u Uj Pjp L2(j )Orthogonal basisTime IntegrationlExplicit Runge-KuttaTVB method of Cockburn and Shu (1989)Local time stepping, Remacle et al. (2002)xtFlux Evaluationl Approximate fn(Uj) by a numerical flux Fn(Uj,Unbj) Define Fn(Uj,Unbj) by a Riemann probleml Possibilities: Upwind: flux from inflow neighbor Lax-Friedrichs: |max| is the maximum absolute eigenvalue of fu Roe: linearized Riemann problem van Leer: flux vector splitting Colella-Woodward: contact surface resolutionLimitingl Limiting: suppress spurious oscillations when p 0 while maintaining orderSlope limiter: Cockburn and Shu (1989)Curvature limiter: Barth (1990)Moment limiter: Biswas et al. (1994)Filtering: Gottlieb et al. (1999)l No robust procedures for multi-dimensional situationsSlope vs. Moment LimitingSlope Limiting Moment LimitingKinematic wave equation: ut + ux = 0p = 2SuperconvergencelOne-dimensional conservation lawlSuperconvergence at Radau pointsAdjerid et al. (1995)Biswas et al. (1994)Superconvergencel Theorem: If p 0, the spatial discretization error of the DG method with Uj Pp on xj-1,xj satisfiesl Proof: Use Galerkin orthogonality, properties of Legendre polynomials, and “strong” superconvergence at downwind element ends cf. Adjerid et al. (2002)A Posteriori Error EstimationlOne-dimensional conservation lawlDG methodlError estimateSuperconvergence in plIf f(u) = au:l ut + aux = 0 p = 08 elementsSuperconvergence in plIf f(u) = au:l ut + aux = 0 p = 18 elementsSuperconvergence in plIf f(u) = au:l ut + aux = 0 p = 28 elementsSuperconvergence in plIf f(u) = au:l ut + aux = 0 p = 38 elementsSuperconvergence in plIf f(u) = au:l ut + aux = 0 p = 48 elementsSolitary Wavesl Nonlinear model:l Exact solution:Solution at t = 1 Effectivity indices at t = 1Two-Dimensional Problemsl Steady linear conservation lawl DG formulation U Pp is the DG solution j- is the inflow boundary of j j+ outflow boundary of jError EstimationlSubtract the exact solution and use the Divergence TheoremlAssume the error has a series expansion h is a mesh parameterError EstimationlUse an “induction” argument to showlOrthogonal basis (on canonical triangle)Error EstimationlShow that2D Radau polynomial?Krivodonova and Flaherty (2001)lStrong superconvergenceA Posteriori Error Estimationl Solve the exit flow and DG problem Complexity is O(p) per elementExamplelConsider = (0,1) x (0,1)Exact solutionError EstimatesN 16 56 160p |e|0 |e|0 |e|0 0 4.85e-2 1.0116 2.49e-4 1.0304 1.49e-2 1.04181 8.27e-4 1.0022 2.16e-4 1.0537 7.85e-5 1.02662 3.11e-5 0.9609 4.16e-6 0.9267 9.24e-7 0.90543 1.71e-6 1.0161 1.04e-7 1.0546 1.47e-8 1.00544 1.07e-7 1.0597 3.32e-9 1.0097 2.8e-10 0.9203SuperconvergenceN 8 32 128p I- I+ I- I