New efficient substepping methods for exponential timestepping
For researchers using exponential integrators on large-scale ODE systems, this work offers a more efficient approach to matrix exponential approximation, though the improvements are incremental.
The paper introduces substepping methods for exponential integrators that recycle Krylov subspaces across substeps to reduce computational cost, and proposes a new second-order integrator using substep information as a corrector. Numerical experiments on a system with 10^6 unknowns show favorable performance compared to existing second-order methods.
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.