On a high-order compact scheme and its utilization in parallel solution of a time-dependent system on a distributed memory processor

Abstract

The focus of this study is the design of a parallel solution method that utilizes a fourth-order compact scheme. The applicability of the method is demonstrated on a time-dependent parabolic system with Neumann boundaries. The core of the parallel computing facilities used in the study is a 2-head-node, 224-compute-node Apple Xserve G5 multiprocessor. The system is first discretized in both time and space such that it remains in its stability regimes, before being solved with the method. The solution requires time marching in which every time step, h t , calls for a single parallel solve of the intermediary subsystems generated. The solution uses p processors ranging in numbers from 3 to 63. The speedups, s p , approach their limiting value of p only when p is small. The solution produces good computational results at large p, but poor results as p becomes progressively small. Also, the parallel solution produces accurate results yielding good speedups and efficiencies only when p is within some reasonable range of values. The intermediary systems generated by this method are linear and fine-grained, therefore, they are best suited for solution on massively-parallel processors. The solution method proposed in this study is, therefore, expected to yield more impressive results if applied in a massively-parallel computing environment.