Implementation of exponential Rosenbrock-type integrators

In this paper, we present a variable step size implementation of exponential Rosenbrock-type methods of orders 2, 3 and 4. These integrators require the evaluation of exponential and related functions of the Jacobian matrix. To this aim, the Real Leja Points Method is used. It is shown that the properties of this method combine well with the particular requirements of Rosenbrock-type integrators. We verify our implementation with some numerical experiments in MATLAB, where we solve semilinear parabolic PDEs in one and two space dimensions. We further present some numerical experiments in FORTRAN, where we compare our method with other methods from literature. We find a great potential of our method for non-normal matrices. Such matrices typically arise in parabolic problems with large advection in combination with moderate diffusion and mildly stiff reactions.