An Additive Multigrid Method Using a Filtering Concept to Solve Flow Simulations on Parallel Computers

Luc Fournier
I. N. R. I. A., France

Stéphane Lanteri
I. N. R. I. A., France

Abstract

Since the 80's, multigrid methods are well known to be caracterized by a relatively low parallel efficiency. Nevertheless several works have shown the great interest in these methods for the numerical simulation of industrial flows. Therefore, it seems still useful to develop multigrid methods adapted to parallel computers. Several approaches have already been investigated.

Here, an additive formulation is analysed in detail. This method has been initially proposed by R. Tuminaro [Tum92]. The originality is in the use of filtering techniques in order to disconnect the different parts of the frequency spectrum of the residual. Each part of the filtered residual is used to create a correction problem which is treated on the most adapted grid. After a detailed presentation of the algorithm, a convergence analysis is conducted using properties introduced in [Hac85]. This analysis shows that the additive method should have a convergence rate independent of the mesh size. \\

This additive multigrid method has been used to solve linear systems involved in compressible fluid flow calculations. The numerical framework is based on the use of unstructured meshes with an agglomeration technique in order to provide coarse grids [LSD92]. We solve the Euler equations for steady flows using a finite volume discretization based on the MUSCL technique. The steady solution is reached using an unsteady formulation with a pseudo-time step. An implicit formulation allows us to define a time step not restricted by a Courant-Friedrichs-Lewy stability condition. So, due to the use of a linearization of the jacobian matrix of the fluxes, each time step results in the resolution of a linear system. The multigrid method is used to accelerate the resolution of this linear system. \\

The method has first been validated through a sequential implementation. Then we have developed a parallel version using a mesh partitioning approach. But, as the convergence rate of the additive formulation is worst than the multiplicative one, we decided to use the additive method only for the coarsest levels where the ratio between communication and calculation is unfavorable. \\

[Hac85] W. Hackbusch. Multi-grid Methods and Applications, volume 4 of Springer series in Computationl Mathematics. Springer Verlag, 1985.

[LSD92] M.-H. Lallemand, H. Steve, and A. Dervieux. Unstructured Multigridding by Volume Agglomeration: Current Status. Computers and Fluids, 21:3.

[Tum92] R. S. Tuminaro. A Highly Parallel Multigrid-like Method for the Solution of the Euler Equations. Sci. Stat. Comput., 13(1):88--100, January 1992.