High-order discontinuous Galerkin methods using an hp-multigrid approach

The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse ''grid'' levels are constructed by reducing the order (p) of approximation of the discretization using hierarchical basis functions (p-multigrid), together with the traditional (h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are treated implicitly (i.e., inverted directly) and all other entries are treated explicitly. The methodology is developed for the two-dimensional non-linear Euler equations on unstructured grids, using both non-linear (FAS) and linear (CGC) multigrid schemes. Results are presented for the channel flow over a bump and a uniform flow over a four element airfoil. Current results demonstrate convergence rates which are independent of both order of accuracy (p) of the discretization and level of mesh resolution (h).

[1]  D. A. Dunavant.  Economical symmetrical quadrature rules for complete polynomials over a square domain , 1985 .

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

[3]  J. V. D. Vegt,et al.  Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics , 2002 .

[4]  I. Babuska,et al.  Finite Element Analysis , 2021 .

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

[6]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[7]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

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

[9]  George Em Karniadakis,et al.  Galerkin and discontinuous Galerkin spectral/hp methods , 1999 .

[10]  Paul F. Fischer,et al.  Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method , 2005, J. Sci. Comput..

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

[12]  Thomas A. Manteuffel,et al.  Algebraic multigrid for higher-order finite elements , 2005 .

[13]  D. Mavriplis.  An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers , 2001 .

[14]  David L. Darmofal,et al.  DEVELOPMENT OF A HIGHER-ORDER SOLVER FOR AERODYNAMIC APPLICATIONS , 2004 .

[15]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

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

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

[18]  Derek M. Causon,et al.  On the Choice of Wavespeeds for the HLLC Riemann Solver , 1997, SIAM J. Sci. Comput..

[19]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[20]  Dimitri J. Mavriplis,et al.  Agglomeration multigrid for two-dimensional viscous flows , 1995 .

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

[22]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[23]  I. M. Hall,et al.  Test cases for the plane potential flow past multi-element aerofoils , 1985, The Aeronautical Journal (1968).

[24]  Bernardo Cockburn,et al.  Discontinuous Galerkin Methods for Convection-Dominated Problems , 1999 .

[25]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

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

[27]  Giuseppe Gambolati,et al.  Nested Iterations for Symmetric Eigenproblems , 1995, SIAM J. Sci. Comput..

[28]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[29]  Dimitri J. Mavriplis,et al.  UNSTRUCTURED MESH GENERATION AND ADAPTIVITY , 1995 .

[30]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

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

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

[33]  Michael A. Leschziner,et al.  Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows , 1997 .