A Comparative Study of Different Reconstruction Schemes for a Reconstructed Discontinuous Galerkin Method on Arbitrary Grids

A comparative study of different reconstruction schemes for a reconstruction-based discontinuous Galerkin, termed RDG(P1P2) method is performed for compressible flow problems on arbitrary grids. The RDG method is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution via a reconstruction scheme commonly used in the finite volume method. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are implemented to obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. These three reconstruction/recovery methods are compared for a variety of compressible flow problems on arbitrary meshes to access their accuracy and robustness. The numerical results demonstrate that all three reconstruction methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy and robustness.

[1]  ZhangLaiping,et al.  A class of hybrid DG/FV methods for conservation laws I , 2012 .

[2]  Hong Luo,et al.  A Discontinuous Galerkin Method Based on a Gas Kinetic Scheme for the Navier-Stokes Equations on Arbitrary Grids , 2009 .

[3]  Claus-Dieter Munz,et al.  A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes , 2007, J. Comput. Phys..

[4]  Michael Dumbser,et al.  Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations , 2009, J. Comput. Phys..

[5]  Rainald Löhner,et al.  Fast p-Multigrid Discontinuous Galerkin Method for Compressible Flows at All Speeds , 2008 .

[6]  S. Rebay,et al.  GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations , 2000 .

[7]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[8]  Hong Luo,et al.  A Parallel Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Aritrary Grids , 2010 .

[9]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[10]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[11]  Rainald Löhner,et al.  A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids , 2008, J. Comput. Phys..

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

[13]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[14]  M. Y. Hussaini,et al.  An efficient implicit discontinuous spectral Galerkin method , 2001 .

[15]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[16]  Bram van Leer,et al.  A Discontinuous Galerkin Method for Diffusion Based on Recovery , 2007 .

[17]  Robert Nourgaliev,et al.  Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for Multiphysics Problems , 2010 .

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

[19]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[20]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[21]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[22]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[23]  Brian T. Helenbrook,et al.  Analysis of ``p''-Multigrid for Continuous and Discontinuous Finite Element Discretizations , 2003 .

[24]  Bram van Leer,et al.  Bilinear Forms for the Recovery-Based Discontinuous Galerkin Method for Diffusion , 2007 .

[25]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .

[26]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[27]  H. T. Huynh,et al.  A Reconstruction Approach to High -Order Schemes Including Discontinuous Galerkin for Diffusion , 2009 .

[28]  Vincent Mousseau,et al.  A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids , 2010, J. Comput. Phys..

[29]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[30]  Harold L. Atkins,et al.  QUADRATURE-FREE IMPLEMENTATION OF DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS , 1996 .

[31]  Rainald Löhner,et al.  On the computation of steady‐state compressible flows using a discontinuous Galerkin method , 2008 .

[32]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[33]  George Em Karniadakis,et al.  A Discontinuous Galerkin Method for the Viscous MHD Equations , 1999 .

[34]  Andreas Dedner,et al.  The compact discontinuous Galerkin method for elliptic problems , 2008 .

[35]  Rainald Löhner,et al.  A Discontinuous Galerkin Method Using Taylor Basis for Compressible Flows on Arbitrary Grids , 2007 .

[36]  Hongwei Liu,et al.  A Runge-Kutta discontinuous Galerkin method for viscous flow equations , 2007, J. Comput. Phys..

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