Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm

In this article, the wave propagation properties of the 1D spectral volume method are studied through analysis of the Fourier footprint of the schemes. A p-multigrid algorithm for the spectral volume method is implemented. Restriction and prolongation operators are discussed and the efficiency of the smoothing operators is analyzed. The results are verified for simple 1D advection problems and for a quasi-1D Euler flow. It is shown that a significant decrease in computational effort is possible with the p-multigrid algorithm.

[1]  Zhi J. Wang,et al.  Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids , 2004 .

[2]  A. Brandt.  Guide to multigrid development , 1982 .

[3]  Zhi J. Wang,et al.  Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems , 2004 .

[4]  Anthony T. Patera,et al.  Spectral element multigrid. I. Formulation and numerical results , 1987 .

[5]  Z. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids , 2002 .

[6]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

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

[8]  Mengping Zhang,et al.  An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods , 2005 .

[9]  M. Y. Hussaini,et al.  An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems , 1999 .

[10]  Francesco Bassi,et al.  Numerical Solution of the Euler Equations with a Multiorder Discontinuous Finite Element Method , 2003 .

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

[12]  Zhi Jian Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids III: One Dimensional Systems and Partition Optimization , 2004, J. Sci. Comput..

[13]  Yuzhi Sun,et al.  Spectral (finite) volume method for conservation laws on unstructured grids VI: Extension to viscous flow , 2006, J. Comput. Phys..

[14]  Chris Lacor,et al.  Optimization of time integration schemes coupled to spatial discretization for use in CAA applications , 2006, J. Comput. Phys..

[15]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[16]  Martine Baelmans,et al.  A finite volume formulation of compact central schemes on arbitrary structured grids , 2004 .

[17]  Marcel Vinokur,et al.  Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems , 2006, J. Comput. Phys..

[18]  B. T. Helenbrook,et al.  Application of “ p ”-multigrid to discontinuous Galerkin formulations of the Poisson equation , 2008 .

[19]  Zhi J. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation , 2002 .

[20]  Krzysztof J. Fidkowski,et al.  A high-order discontinuous Galerkin multigrid solver for aerodynamic applications , 2004 .