  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.
