Time stepping algorithm with spatial parallelisation is commonly used to solve time dependent partial differential equations. Computation in each time step is carried out using all processors available before sequentially advancing to the next time step. In cases where few spatial components are involved and there are relatively many processors available for use, this will result in fine granularity and decreased scalability. Naturally one alternative is to parallelise the temporal domain. Several time parallelisation algorithms have been suggested for the past two decades. One of them is the pipelined iterations across time steps. In this pipelined time stepping method, communication however is extensive between time steps during the pipelining process. This causes a decrease in performance on distributed memory environment which often has high message latency. We present a modified pipelined time stepping algorithm based on delayed pipelining and reduced communication strategies to improve overall execution time on a distributed memory environment using MPI. Our goal is to reduce the inter-time step communications while providing adequate information for the next time step to converge. Numerical result confirms that the improved algorithm is faster than the original pipelined algorithm and sequential time stepping algorithm with spatial parallelisation alone. The improved algorithm is most beneficial for fine granularity time dependent problems with limited spatial parallelisation.
Combining iterative methods with appropriate preconditioners is a worthwhile effort in improving the performances of the methods since the reliability of these methods have been shown to improve by the use of appropriate preconditioning techniques. However, the hardest issue is to find the suitable preconditioners which are computationally inexpensive and easy to solve for the group methods. The aim of this paper is to study the performance of the Explicit Group (EG) method (Evans & Yousif 1986) preconditioned by a specific ‘splitting’ approach in solving the two dimensional elliptic partial differential equation. Our goal in this work is to investigate whether the performance of this group method is affected by this preconditioner. The experimental work performed is reported and discussed.