Application of the p-Multigrid Algorithm to Discontinuous Galerkin Formulations of the Compressible Euler Equation

Abstract

p-multigrid is an iterative algorithm used to obtain solutions to hp-finite element discretizations. In this work, we have investigated the use of p-multigrid (for p = 2 approximations) and agglomeration multigrid (for p = 0 approximations) to obtain the solution to the discontinuous Galerkin formulation of the compressible Euler equations applied to a model problem. Various diagonal, line solve and sweeping relaxation schemes have been investigated to compare their performance. The model problem analyzed is a 2-D free-steam subsonic flow which is periodic in one direction. For this problem, we have used a structured rectangular mesh and linear analysis is used to obtain the convergence rates (damping factors) of the different schemes. For all orders of p, the damping factors are observed to vary significantly with the angle of the flow with respect to the mesh. The block line relaxation schemes including the alternate direction line (ADL) scheme, with an under-relaxation factor of 2/3, showed the least variation with flow angle. Even when the flow is perfectly aligned with the mesh, these schemes converge well unlike the diagonal relaxation schemes. We also studied the variation of the damping factors with grid refinement and here again the line relaxation schemes showed grid independence at all flow ∗Ph.D. candidate, Mechanical and Aeronautical Engineering Department, mascarbs@clarkson.edu, AIAA Student Member. †Associate Professor, Mechanical and Aeronautical Engineering Department, helenbrk@clarkson.edu, AIAA Member. ‡Senior Research Scientist, Computational Modeling and Simulation Branch, h.l.atkins@larc.nasa.gov, AIAA member 1 18th AIAA Computational Fluid Dynamics Conference 25 28 June 2007, Miami, FL AIAA 2007-4331 Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. angles, whereas the diagonal schemes were only grid independent at flow angles away from 0◦ and 180◦. For the p = 0 approximations, the block line solve scheme with symmetric Gauss-Seidel (SGS) sweeps and two-level agglomeration multigrid, performs the best, giving damping factors in the range of 0.35 to 0.8 for a wide range of flow angles. The ADL multigrid scheme with GaussSeidel sweep also performs very well when the GS sweep is in the direction of the x-component of the flow velocity, giving damping factors between 0.33 and 0.42 for all flow angles less than 90◦. For the p = 2 approximations, the ADL relaxation coupled with a two level p-multigrid algorithm (p coarsened from 2 to 1) has the best convergence properties with damping factors in the range of 0.65 to 0.80 for all flow angles other than those between 80◦ and 100◦. When p is coarsened from 1 to 0 however, the p-multigrid algorithm does not perform well. We also explain this anomalous behavior of the p-multigrid algorithm.