Uniform Approximation of $\varphi$-Functions in Exponential Integrators by a Rational Krylov Subspace Method with Simple Poles

Abstract

We consider the approximation of the matrix $\varphi$-functions that appear in exponential integrators for stiff systems of differential equations. For stiff systems, the field-of-values of the occurring matrices is large and lies somewhere in the left complex half-plane. In order to obtain an efficient method uniformly for all matrices with a field-of-values in the left complex half-plane, we consider the approximation by a rational Krylov subspace method with equidistant poles of order one on the line $\mbox{Re}\,z = \gamma > 0$. We present error bounds that predict a faster convergence rate as for the resolvent Krylov subspace approximation using a single repeated pole at $\gamma > 0$. Poles of order one allow moreover for a parallel implementation of the corresponding rational Krylov subspace decomposition. We analyze the convergence of the proposed rational Krylov subspace method and present numerical experiments that illustrate our results.