New efficient substepping methods for exponential timestepping

TL;DR

This paper introduces efficient substepping methods for exponential integrators that recycle Krylov subspaces to reduce computational costs and proposes a new second-order scheme with improved accuracy for large systems.

Contribution

It develops and analyzes substepping schemes with Krylov recycling and introduces a novel second-order integrator leveraging substep information for enhanced performance.

Findings

Recycling Krylov subspaces reduces computational costs.

The new second-order integrator outperforms existing methods.

Applicable to large systems with up to 10^6 unknowns.

Abstract

Exponential integrators are time stepping schemes which exactly solve the linear part of a semilinear ODE system. This class of schemes requires the approxima- tion of a matrix exponential in every step, and one successful modern method is the Krylov subspace projection method. We investigate the effect of breaking down a single timestep into arbitrary multiple substeps, recycling the Krylov subspace to minimise costs. For these recyling based schemes we analyse the lo- cal error, investigate them numerically and show they can be applied to a large system with 106 unknowns. We also propose a new second order integrator that is found using the extra information from the substeps to form a corrector to increase the overall order of the scheme. This scheme is seen to compare favourably with other order two integrators.