A p-Multigrid spectral difference method for viscous compressible flow using 2D quadrilateral meshes

The work focuses on the development of a 2D quadrilateral element based Spectral Difference solver for viscous flow calculations, and the application of the p-multigrid method and implicit time-stepping to accelerate convergence. This paper extends the previous work by Liang et al (2009) on the p-multigrid method for 2D inviscid compressible flow, to viscous flows. The high-order spectral difference solver for unstructured quadrilateral meshes is based on the formulation of Sun et al for unstructured hexahedral elements. The p-multigrid method operates on a sequence of solution approximations of different polynomial orders ranging from one upto four. An efficient preconditioned Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit scheme is also implemented. The spectral difference method is applied to a variety of inviscid and viscous compressible flow problems. The speed-up using the p-Multigrid and Implicit time-stepping techniques is also demonstrated.

[1]  Chunlei Liang,et al.  A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids , 2009 .

[2]  Antony Jameson,et al.  Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations , 2007, J. Sci. Comput..

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

[4]  Chunlei Liang,et al.  A p-Multigrid Spectral Difference method with explicit and implicit smoothers on unstructured grids , 2007 .

[5]  Zhi J. Wang,et al.  Efficient Implicit LU-SGS Algorithm for High-Order Spectral Difference Method on Unstructured Hexahedral Grids , 2007 .

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

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

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

[9]  Zhi J. Wang,et al.  High-Order Multidomain Spectral Difference Method for the Navier-Stokes Equations , 2006 .

[10]  Zhi J. Wang,et al.  Extension of the spectral volume method to high-order boundary representation , 2006 .

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

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

[13]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..

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

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

[16]  Y. Maday,et al.  Spectral element multigrid. Part 2: Theoretical justification , 1988 .

[17]  A. Jameson,et al.  Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations , 1988 .

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

[19]  Antony Jameson,et al.  Lower-upper implicit schemes with multiple grids for the Euler equations , 1987 .