New efficient substepping methods for exponential timestepping

Exponential integrators are time stepping schemes which exactly solvethe linear part of a semilinear ODE system. This class of schemesrequires the approximation of a matrix exponential in every step, andone successful modern method is the Krylov subspace projectionmethod. We investigate the effect ofbreaking down a single timestep into arbitrary multiple substeps,recycling the Krylov subspace to minimise costs. For these recycling based schemes we analyse the local error, investigate them numerically and show they can be applied to a large system with $10^6$ unknowns.We also propose a new second order integrator that is found using the extrainformation from the substeps to form a corrector to increase theoverall order of the scheme. This scheme is seen to compare favorablywith other order two integrators.