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A simple shock-capturing technique for high-order discontinuous Galerkin methods. (English) Zbl 1253.76058
Summary: We present a novel shock-capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high-order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high-order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76N99 Compressible fluids and gas dynamics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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References:
[1] Bassi, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics 131 (2) pp 267– (1997) · Zbl 0871.76040
[2] Cockburn, TVB Runge-Kutta local projection Discontinuous Galerkin finite element method for conservation laws. II. General framework, Mathematics of Computation 52 (186) pp 411– (1989)
[3] Biswas, Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics 14 (1-3) pp 255– (1994) · Zbl 0826.65084
[4] Cockburn, The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis 35 (6) pp 2440– (1998) · Zbl 0927.65118
[5] Cockburn, Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing 16 (3) pp 173– (2001) · Zbl 1065.76135
[6] Krivodonova, Shock detection and limiting with Discontinuous Galerkin methods for hyperbolic conservation laws, Applied Numerical Mathematics 48 (3-4) pp 323– (2004) · Zbl 1038.65096
[7] Remacle, Anisotropic adaptive simulation of transient Flows using Discontinuous Galerkin methods, International Journal for Numerical Methods in Engineering 62 (7) pp 899– (2005) · Zbl 1078.76042
[8] Nguyen, An implicit high-order hybridizable Discontinuous Galerkin method for nonlinear convection-diffusion equations, Journal of Computational Physics 228 (23) pp 8841– (2009) · Zbl 1177.65150
[9] Barth TJ 1995
[10] LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics (2002) · Zbl 1010.65040
[11] Von Neumann, A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics 21 pp 232– (1950) · Zbl 0037.12002
[12] Persson P-O Peraire J Sub-cell shock capturing for Discontinuous Galerkin methods Proc. of the 45th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada 2006
[13] Barter, Shock capturing with PDE-based artificial viscosity for DGFEM:Part I. Formulation, Journal of Computational Physics 229 (5) pp 1810– (2010) · Zbl 1329.76153
[14] Premasuthan S Liang C Jameson A Computation of flow with shocks using spectral difference scheme with artificial viscosity Proc. of the 48th AIAA CAerospace Sciences Meeting including the New horizons Forum and Aerospace Esposition Orlando, FL 2010
[15] Casoni, One-Dimensional Shock-Capturing for High-Order Discontinuous Galerkin Methods, ECCOMAS Multidisciplinary Jubilee Symposium 14 (2009) · Zbl 1189.76335
[16] LeVeque, Numerical methods for conservation laws (1992)
[17] Cockburn, TVB Runge-Kutta local projection Discontinuous Galerkin finite element method for conservation laws. III. one-dimensional systems, Journal of Computational Physics 84 (1) pp 90– (1989) · Zbl 0677.65093
[18] Shu, Efficient implementation of essentially nonoscillatory shock-capturing schemes, Journal of Computational Physics 77 (2) pp 439– (1988) · Zbl 0653.65072
[19] Qiu, Runge-Kutta Discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (3) pp 907– (2005) · Zbl 1077.65109
[20] Zhu, Runge-Kutta Discontinuous Galerkin method using WENO limiters II: Unstructured meshes, Journal of Computational Physics 227 (9) pp 4330– (2008) · Zbl 1157.65453
[21] Donea, Finite element methods for flow problems (2003)
[22] Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics 135 (2) pp 250– (1997) · Zbl 0890.65094
[23] Hirsch, Numerical Computation of Internal and External Flows: Computational methods for inviscid and viscous flows (1990) · Zbl 0742.76001
[24] Chavent, The local projection P0P1-Discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modélisation Mathématique et Analyse Numérique 23 (4) pp 565– (1989)
[25] Van Leer, Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method, Journal of Computational Physics 135 (2) pp 229– (1997) · Zbl 0939.76063
[26] Cueto-Felgueroso, High-order Finite Volume methods and multiresolution reproducing kernels, Archives of Computational Methods in Engineering 15 (2) pp 185– (2008) · Zbl 1300.76018
[27] Nguyen NC Persson P-O Peraire J RANS solutions using high order Discontinuous Galerkin methods Proc. of the 44th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada 2007
[28] Barth TJ Jespersen DC The design and application of upwind schemes on unstructured meshes Proc. of the 27th AIAA Aerospace Sciences Meeting Reno, NV 1989
[29] Venkatakrishnan, Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, Journal of Computational Physics 118 (1) pp 120– (1995) · Zbl 0858.76058
[30] Krivodonova, Limiters for high-order Discontinuous Galerkin methods, Journal of Computational Physics 226 (1) pp 879– (2007) · Zbl 1125.65091
[31] Koornwinder, Askey-Wilson polynomials for root systems of type BC, Contemporary Mathematics 138 pp 189– (1992) · Zbl 0797.33014
[32] Mavriplis, Adaptive mesh strategies for the spectral element method, Computer Methods in Applied Mechanics and Engineering 116 (1-4) pp 77– (1994) · Zbl 0826.76070
[33] Gottlieb, Spectral methods for hyperbolic problems, Journal of Computational and Applied Mathematics 128 (1-2) pp 83– (2001) · Zbl 0974.65093
[34] Sevilla, NURBS-enhanced finite element method (NEFEM), International Journal for Numerical Methods in Engineering 76 (1) pp 56– (2008) · Zbl 1162.65389
[35] Sod, survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational Physics 27 (1) pp 1– (1978) · Zbl 0387.76063
[36] Burbeau, A problem-independent limiter for high-order Runge-Kutta Discontinuous Galerkin methods, Journal of Computational Physics 169 (1) pp 111– (2001) · Zbl 0979.65081
[37] Casoni, Un método de captura de choques basado en las funciones de forma para Galerkin discontinuo en alto orden, to appear in Revista Internacional de Meétodos Numéricos en Ingeniería (2011)
[38] Moukalled, A high-resolution pressure-based algorithm for fluid flow at all speeds, Journal of Computational Physics 168 (1) pp 101– (2001) · Zbl 0991.76047
[39] Luo, A Hermite WENO-based limiter for Discontinuous Galerkin method on unstructured grids, Journal of Computational Physics 225 (1) pp 686– (2007) · Zbl 1122.65089
[40] Dolejsi, A semi-implicit Discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, Journal of Computational Physics 198 (2) pp 727– (2004) · Zbl 1116.76386
[41] Anderson, Modern Compressible Flow (1982)
[42] Demirdžić, A collocated Finite Volume method for predicting flows at all speeds, International Journal for Numerical Methods in Fluids 16 (12) pp 1029– (1993) · Zbl 0774.76066
[43] Hartmann, Adaptive Discontinuous Galerkin finite element methods for the compressible Euler equations, Journal of Computational Physics 183 (2) pp 508– (2002) · Zbl 1057.76033
[44] Emery, An evaluation of several differencing methods for inviscid fluid flow problems, Journal of Computational Physics 2 (3) pp 306– (1968) · Zbl 0155.21102
[45] Woodward, The numerical-simulation of two-dimensional fluid-flow with strong shocks, Journal of Computational Physics 54 (1) pp 115– (1984) · Zbl 0573.76057
[46] Holden, An unconditionally stable method for the Euler equations, Journal of Computational Physics 150 (1) pp 76– (1999) · Zbl 0922.76251
[47] Xu, A conservation constrained Runge-Kutta Discontinous Galerkin method with the improved CFL condition for conservation laws, submitted to SIAM Journal on Scientific Computing (2011)
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