Parallel iterative linear solvers for multistep Runge-Kutta methods

This paper deals with solving stiff systems of differential equations by implicit Multistep Runge-Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge-Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. With a parallel iterative linear system solver, especially designed for MRK methods, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-steps-stage MRK applied with this technique is on s processors equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. A simple implementation of both methods shows that, for the same number of Newton iterations, the accuracy delivered by the new method is higher than that of BDF.