Variable step implementation of ETD methods for semilinear problems

Spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations. Exponential integrators solve exactly the linear part, which is assumed to be responsible of the stiffness of the problem, while the nonlinear terms are approximated numerically.In this paper, a variable step implementation of the exponential time differencing methods of multistep type is considered. A local error estimate of exponential type, whose computation is for free, is introduced and some implementation issues for diagonalizable matrices are addressed. Numerical results showing the behaviour of the proposed variable step implementation are provided too.