Multigrid Methods for Petroleum Reservoir Simulation on SIMD Machines

Abstract In this paper we consider several multigrid methods on SIMD (single instruction, multiple data) architectures. The advantages and disadvantages of these methods are discussed. Actual timing data will be presented only at the meeting, since otherwise this paper would contain data five months old, rendering it obsolete in the current fast-moving world of parallel computation. In a previous paper [DMRRS], a semicoarsening multigrid procedure was considered for solving three-dimensional problems that arise from pressure equations in reservoir simulation. In this procedure, semicoarsening was coupled with plane relaxation in order to accommodate anisotropies and heterogeneities. Subsequent work by Schaffer [S,SW] has improved the robustness of this procedure. Results of the implementation on the CM-200 of the twodimensional version of this procedure will be described, as will progress on the implementation on the CM-5 of the two-dimensional and three-dimensional versions. Another possibility involves the use of Mulder's method as developed by Naik and Van Rosendale. [NR] In this method semicoarsened grids in both x and y (and z in three dimensions) are employed. The advantage is that general anisotropic problems require only point relaxation. This approach is compared to the above approach and shown to have better convergence factors but to be less efficient than the above method on machines like the CRAY-YMP. This method is being implemented on the CM-200 and CM-5 by using concurrent relaxation, interpolation, and restriction. This approach yields great parallelism but at the sacrifice of the splendid convergence factors. A third possibility involves standard coarsening multigrid. This method seems less attractive than semicoarsening multigrid for SIMD machines, especially for three-dimensional problems. A final possibility is the method referred to as BHT in [DIR]. This method uses standard coarsening but for every grid employs four coarse grids, thus keeping all the processors busy at every level. Busyness of processors is not innately virtuous, but the employment of supplemental coarse grids gives rise to a method which requires only point relaxation for simple anisotropic problems. It is the latter feature which makes the BHT method potentially attractive.