Experiences with sparse matrix solvers in parallel ODE software

The use of implicit methods for numerically solving stiff systems of differential equations requires the solution of systems of non-linear equations. Normally these are solved by a Newton-type process, in which we have to solve systems of linear equations. The Jacobian of the derivative function determines the structure of the matrices of these linear systems. Since it often occurs that the components of the derivative function only depend on a small number of variables, the system can be considerably sparse. Hence, it can be worth the effort to use a sparse matrix solver instead of a dense LU-decomposition. This paper reports on experiences with the direct sparse matrix solvers MA28 by Duff, Y12M by Zlatev et al. and one special-purpose matrix solver, all embedded in the parallel ODE solver PSODE by Sommeijer.