A Reconstruction Approach to High -Order Schemes Including Discontinuous Galerkin for Diffusion

We in troduce a new approach to high -order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique . The approach has the advantages of simplicity and economy. It results in several new high -order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient —quantities at each interface shared by the two adjacent cells . In addition, the new approach clarifie s the relation s among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting scheme s. Extensions to the case of quadrilateral meshes ar e obtained via tensor products. For the two -point boundary value problem ( steady state), it is shown that these schemes , which include most popular DG methods, yield ex act common interface quantities as well as exact cell average solutions for nearly all cases. Keywords . Discontinuous Galerkin meth ods, high -order methods, reconstruction, diffusion equation.

[1]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[2]  Chris Lacor,et al.  An accuracy and stability study of the 2D spectral volume method , 2007, J. Comput. Phys..

[3]  F. Brezzi,et al.  Discontinuous Galerkin approximations for elliptic problems , 2000 .

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

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

[6]  Francesco Bassi,et al.  A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows , 2000 .

[7]  Per-Olof Persson,et al.  The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems , 2007, SIAM J. Sci. Comput..

[8]  Marcel Vinokur,et al.  Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids , 2004 .

[9]  F. B. Hildebrand,et al.  Introduction To Numerical Analysis , 1957 .

[10]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[11]  Marcel Vinokur,et al.  Spectral difference method for unstructured grids I: Basic formulation , 2006, J. Comput. Phys..

[12]  J. Z. Zhu,et al.  The finite element method , 1977 .

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

[14]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[15]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[16]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

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

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

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

[20]  Carl De Boor,et al.  Mathematical aspects of finite elements in partial differential equations : proceedings of a symposium conducted by the Mathematics Research Center, the University of Wisconsin--Madison, April 1-3, 1974 , 1974 .

[21]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .

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

[23]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[24]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

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

[26]  David A. Kopriva,et al.  A Staggered-Grid Multidomain Spectral Method for the Compressible Navier-Stokes Equations , 1998 .