Stencils and Problem Partitionings: Their Influence on the Performance of Multiple Processor Systems

Given a discretization stencil, partitioning the problem domain is an important first step for the efficient solution of partial differential equations on multiple processor systems. We derive partitions that minimize interprocessor communication when the number of processors is known a priori and each domain partition is assigned to a different processor. Our partitioning technique uses the stencil structure to select appropriate partition shapes. For square problem domains, we show that nonstandard partitions (e.g., hexagons) are frequently preferable to the standard square partitions for a variety of commonly used stencils. We conclude with a formalization of the relationship between partition shape, stencil structure, and architecture, allowing selection of optimal partitions for a variety of parallel systems.