On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods

We examine four nodal versions of tensor product discontinuous Galerkin spectral element approximations to systems of conservation laws for quadrilateral or hexahedral meshes. They arise from the two choices of Gauss or Gauss-Lobatto quadrature and integrate by parts once (I) or twice (II) formulations of the discontinuous Galerkin method. We show that the two formulations are in fact algebraically equivalent with either Gauss or Gauss-Lobatto quadratures when global polynomial interpolations are used to approximate the solutions and fluxes within the elements. Numerical experiments confirm the equivalence of the approximations and indicate that using Gauss quadrature with integration by parts once is the most efficient of the four approximations.

[1]  Kelly Black,et al.  Spectral element approximation of convection—diffusion type problems , 2000 .

[2]  Gaston H. Gonnet,et al.  Scientific Computation , 2009 .

[3]  M. Y. Hussaini,et al.  Discontinuous Spectral Element Approximation of Maxwell’s Equations , 2000 .

[4]  Davis A. Kopriva,et al.  Computation of electromagnetic scattering with a non‐conforming discontinuous spectral element method , 2002 .

[5]  Marc Duruflé,et al.  Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic , 2009 .

[6]  George Em Karniadakis,et al.  De-aliasing on non-uniform grids: algorithms and applications , 2003 .

[7]  Jan S. Hesthaven,et al.  Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations , 2002 .

[8]  Francis X. Giraldo,et al.  A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases , 2008, J. Comput. Phys..

[9]  F. Farassat,et al.  Aircraft Engine Noise Scattering By Fuselage and Wings: A Computational Approach , 2003 .

[10]  David Jon Furbish,et al.  Numerical Solution of the Dam-Break Problem with a Discontinuous Galerkin Method , 2004 .

[11]  Francis X. Giraldo,et al.  A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Modeling , 2009, SIAM J. Sci. Comput..

[12]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers , 2009 .

[13]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

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

[15]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[16]  M. Y. Hussaini,et al.  Discontinuous Spectral Element Solution of Acoustic Radiation from Thin Airfoils , 2001 .

[17]  F. Farassat,et al.  Aircraft Engine Noise Scattering - A Discontinuous Spectral Element Approach , 2002 .

[18]  M. Y. Hussaini,et al.  Computation of Engine Noise Propagation and Scattering off An Aircraft , 2002 .

[19]  David Jon Furbish,et al.  Application of the discontinuous spectral Galerkin method to groundwater flow , 2004 .

[20]  Wei Cai,et al.  Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides. , 2004, Optics express.

[21]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations , 2009 .

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

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

[24]  Kelly Black,et al.  A conservative spectral element method for the approximation of compressible fluid flow , 1999, Kybernetika.

[25]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[26]  David A. Kopriva,et al.  Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes , 2006, J. Sci. Comput..

[27]  George Em Karniadakis,et al.  Regular Article: A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains , 1999 .

[28]  Claudio Canuto,et al.  Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) , 2007 .

[29]  Shaozhong Deng,et al.  Numerical simulation of optical coupling and light propagation in coupled optical resonators with size disorder , 2007 .