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. Like in [HS96], where the one-step RK methods were considered, 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-step s-stage MRK applied with PILSMRK on s processors is equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. If both methods perform the same number of Newton iterations, then the accuracy delivered by the new method is also higher than that of BDF.

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Department of Numerical Mathematics [NM]
Numerical mathematics

Messina, E, de Swart, J.J.B, & van der Veen, W.A. (1996). Parallel iterative linear solvers for multistep Runge-Kutta methods. Department of Numerical Mathematics [NM]. CWI.