Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs

We report on our efforts towards the design of efficient parallel hierarchical iterative methods for the solution of sparse and irregularly structured linear systems resulting from CFD applications. The solution strategies considered here share a central numerical kernel which consists in a linear multigrid by volume agglomeration method. Starting from this method, we study two parallel solution strategies. The first variant results from a direct intra-grid parallelization of multigrid operations on coarse grids. The second variant is based on an additive Schwarz domain decomposition algorithm which is formulated at the continuous level through the introduction of specific interface conditions. In this variant, the linear multigrid by volume agglomeration method is used to approximately solve the local systems obtained at each iteration of the Schwarz algorithm. As a result, the proposed hybrid domain decomposition/multigrid method can be viewed as a particular form of parallel multigrid in which multigrid acceleration is applied on a subdomain basis, these local calculations being coordinated by an appropriate domain decomposition iteration at the global level. The parallel performances of these two parallel multigrid methods are evaluated through numerical experiments that are performed on several clusters of PCs with different computational nodes and interconnection networks.

[1]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[2]  Stéphane Lanteri,et al.  A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes , 2001 .

[3]  Stéphane Lanteri,et al.  Parallel Solutions of Compressible Flows Using Overlapping and Non-Overlapping Mesh Partitioning Strategies , 1996, Parallel Comput..

[4]  Dimitri J. Mavriplis,et al.  Directional agglomeration multigrid techniques for high Reynolds number viscous flow solvers , 1998 .

[5]  Dimitri J. Mavriplis,et al.  A 3D AGGLOMERATION MULTIGRID SOLVER FOR THE REYNOLDS-AVERAGED NAVIER-STOKES EQUATIONS ON UNSTRUCTURED MESHES , 1995 .

[6]  Valeria Simoncini,et al.  Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing , 2003, SIAM J. Sci. Comput..

[7]  Alain Dervieux,et al.  Unstructured multigridding by volume agglomeration: Current status , 1992 .

[8]  Gilles Carré,et al.  An implicit multigrid method by agglomeration applied to turbulent flows , 1997 .

[9]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[10]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[11]  V. Frayssé,et al.  A relaxation strategy for inexact matrix-vector products for Krylov methods , 2000 .

[12]  Stéphane Lanteri,et al.  Parallel linear multigrid by agglomeration for the acceleration of 3D compressible flow calculations on unstructured meshes , 2000, Numerical Algorithms.

[13]  Stéphane Lanteri,et al.  Parallel linear multigrid algorithms for the acceleration of compressible flow calculations , 2000 .

[14]  Alain Dervieux,et al.  A SEMI-COARSENING STRATEGY FOR UNSTRUCTURED MULTIGRID BASED ON AGGLOMERATION , 1998 .

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

[16]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[17]  William Gropp,et al.  High-performance parallel implicit CFD , 2001, Parallel Comput..

[18]  Alfio Quarteroni,et al.  HOMOGENEOUS AND HETEROGENEOUS DOMAIN DECOMPOSITION FOR COMPRESSIBLE FLUID FLOWS AT HIGH REYNOLDS NUMBERS , 1998 .

[19]  L. Fezoui,et al.  A class of implicit upwind schemes for Euler simulations with unstructured meshes , 1989 .

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

[21]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[22]  Hervé Guillard,et al.  A Second Order Defect Correction Scheme for Unsteady Problems , 1996 .

[23]  Stéphane Lanteri,et al.  Parallel Linear Multigrid Algorithms Applied to the Acceleration of Compressible Flows , 1998 .