A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

This is the accepted version of the following article: [Huerta, A., Casoni, E. and Peraire, J. (2012), A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int. J. Numer. Meth. Fluids, 69: 1614–1632. doi:10.1002/fld.2654], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.2654/full

[1]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[2]  Per-Olof Persson,et al.  RANS Solutions Using High Order Discontinuous Galerkin Methods , 2007 .

[3]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[4]  Antonio Huerta,et al.  3D NURBS‐enhanced finite element method (NEFEM) , 2008 .

[5]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[6]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[7]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[8]  J. Remacle,et al.  Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws , 2004 .

[9]  ZhuJun,et al.  Runge-Kutta discontinuous Galerkin method using WENO limiters II , 2005 .

[10]  Chunlei Liang,et al.  Computation Of Flows with Shocks Using Spectral Dierence Scheme with Articial Viscosity , 2010 .

[11]  M. Perić,et al.  A collocated finite volume method for predicting flows at all speeds , 1993 .

[12]  D. Gottlieb,et al.  Spectral methods for hyperbolic problems , 2001 .

[13]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations , 2009, J. Comput. Phys..

[14]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[15]  Lilia Krivodonova,et al.  Limiters for high-order discontinuous Galerkin methods , 2007, J. Comput. Phys..

[16]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[17]  Knut-Andreas Lie,et al.  An Unconditionally Stable Method for the Euler Equations , 1999 .

[18]  Ignasi Colominas,et al.  High-order Finite Volume Methods and Multiresolution Reproducing Kernels , 2008 .

[19]  David L. Darmofal,et al.  Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation , 2010, J. Comput. Phys..

[20]  Ashley F. Emery,et al.  An Evaluation of Several Differencing Methods for Inviscid Fluid Flow Problems , 1968 .

[21]  M. Darwish,et al.  A high-resolution pressure-based algorithm for fluid flow at all speeds , 2001 .

[22]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[23]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[24]  Vít Dolejší,et al.  A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow , 2004 .

[25]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[26]  Catherine Mavriplis,et al.  Adaptive mesh strategies for the spectral element method , 1992 .

[27]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[28]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[29]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[30]  Rainald Löhner,et al.  A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids , 2007, J. Comput. Phys..

[31]  Pierre Sagaut,et al.  A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods , 2001 .

[32]  R. Hartmann,et al.  Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations , 2002 .

[33]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[34]  J. Remacle,et al.  Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods , 2005 .

[35]  V. Venkatakrishnan.  Convergence to steady state solutions of the Euler equations on unstructured grids with limiters , 1995 .

[36]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .