Key to the development of an efficient parallel reservoir simulator is the implementation of an efficient parallel linear equation solver. Although recent advances in distributed memory parallel computers have seen a significant improvement in communications overhead among the processors, increases in processor performance have kept the ratio of communications to node performance about the same. Previous published parallel linear equation solvers have emphasized the importance of reduced communications and increased granularity for the solver partitioning among the processors. This study investigates the implementation of asynchronous message-passing schemes to further reduce communications overhead for the parallel solver. The basic idea is for each processor to proceed with the linear solution without interruption from waiting to receive messages. If data is available in the buffer from other processors, however, it is utilized.

This study implements an asynchronous parallel preconditioner for ORTHOMIN(k). The preconditioner has two parts: Z-line relaxation and a block incomplete LU factorization (ILU(O)). Also included in the solution is the use of a reduced three-dimensional subproblem which is solved on one of the processors. Because of the asynchronous implementation of the solver, each of the nodes may perform a different number of iterations and converge to different tolerances; however, the overall tolerance for convergence is checked periodically to determine when the linear solution can be halted. This technique has been implemented on the IBM SP2, Intel IPSC/860 and Intel Paragon with problem sizes varying from 4096 to 184,320 grid blocks. For the largest problem significant performance improvements were observed.

For the range of problems investigated the linear solution technique showed excellent robustness and efficiencies. Number of iterations increased only slightly with increasing number of nodes. Moreover, speedups of up to 26 and 37 were observed for 32 and 64 nodes, respectively, for the largest problem investigated.

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