Execution Behavior Analysis of Parallel Schemes for Implicit Solution Methods for ODEs

In this article, we consider diagonal-implicitly iterated Runge-Kutta (DIIRK) methods for the numerical solution of stiff ordinary differential equations (ODEs) and investigate their performance behavior on a modern cluster system using MPI. DIIRK methods are implicit methods and require the solution of non-linear equation systems in each iteration step. In particular, we are interested in the parallel execution behavior when using different basis Newton methods for solving the resulting non-linear equation systems of different versions of the DIIRK method. We explore the use of direct solution methods based on LU factorization for the resulting linear equation systems as well as the use of Krylov subspace methods and investigate the resulting performance and accuracy.