Diagonal-Implicitly Iterated Runge-Kutta Methods on Distributed Memory Machines

We consider diagonal-implicitly iterated Runge–Kutta methods which are one-step methods for stiff ordinary differential equations providing embedded solutions for stepsize control. In these methods, algorithmic parallelism is introduced at the expense of additional computations. In this paper, we concentrate on the algorithmic structure of these Runge–Kutta methods and consider several parallel variants of the method exploiting algorithmic and data parallelism in different ways. Our aim is to investigate whether these variants lead to good performance on current distributed memory machines such as the Intel Paragon and the IBM SP2. As test application we use ordinary differential equations with dense and sparse right-hand side functions.