Exponential-Krylov methods for ordinary differential equations

Abstract

This paper develops a new class of exponential-type integrators where all the matrix exponentiations are performed in a single Krylov space of low dimension. The new family, called Lightly Implicit Krylov-Exponential (LIKE), is well suited for solving large scale systems of ODEs or semi-discrete PDEs. The time discretization and the Krylov space approximation are treated as a single computational process, and the Krylov space properties are an integral part of the new LIKE order condition theory developed herein. Consequently, LIKE methods require a small number of basis vectors determined solely by the temporal order of accuracy. The subspace size is independent of the ODE under consideration, and there is no need to monitor the errors in linear system solutions at each stage. Numerical results illustrate the favorable properties of new family of methods.