Diagonal - implicity iterated Runge-Kutta methods on distributed memory multiprocessors

We investigate the parallel implementation of the diagonal-implicitly iterated Ruge-Kutta (DIIRK) method, an iteration method based on a predictor-corrector scheme. This method is appropriate for the solution of stiff systems of ordinary differential equations (ODEs) and provides embedded formulae to control the stepsize. We discuss different strategies for the implementation of the DIIRK method on distributed memory multiprocessors which mainly differ in the order of independent computations and the data distribution. In particular, we consider a consecutive implementation that executes the steps of each corrector iteration in sequential order and distributes the resulting equation systems among all available processors, and a group implementation that executes the steps in parallel by independent groups of processors. The performance of these implementations depends on the right hand side of the ODE system: For sparse functions, the group implementations is superior and achieves medium range seedup values. For dense functions, the consecutive implementation is better and achieves good speedup values.

The parallel performance of standard parabolic marching schemes

We compare standard parallel algorithms for solving linear parabolic partial differential equations. The comparison is based on the combined effect of their numerical properties and their parallel performance. We discuss the classical explicit methods (forward Euler, Heun and DuFort-Frankel), the standard implicit methods (BDF1, BDF2 and Crank-Nicolson), the line Hopscotch technique and the ADI formula of McKee and Mitchell. Timing results obtained on a 16-processor Intel hypercube are given. It is shown that parallelism does not alter the ranking of the methods unless the number of grid points per processor is very small.status: publishe