Time-Exact Solution of Large Linear ODE Systems by Block Krylov Subspace Projections

We propose a time-exact Krylov-subspace-based method for solving large linear inhomogeneous systems of ODE (ordinary differential equations). The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the inhomogeneous source term, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method for the matrix exponential. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations. This conference paper is based on the preprint (Botchev, A block Krylov subspace time-exact solution method for linear ODE systems, Memorandum 1973, Department of Applied Mathematics, University of Twente, Enschede, 2012, http://​eprints.​eemcs.​utwente.​nl/​21277/​).