Exponential Rosenbrock methods of order five - construction, analysis and numerical comparisons

The aim of this paper is to construct high-order exponential Rosenbrock methods and to analyze their convergence properties for the time discretization of large-scale systems of stiff differential equations. We present a new and simple approach for deriving the stiff order conditions. These conditions allow us to construct new pairs of embedded methods of high order. As an example, we present a fifth-order method with five stages. For particular problems the order conditions can be simplified. It is then even possible to construct a method of order 5 with three stages only. The error analysis is performed in an abstract framework of strongly continuous semigroups that allows us to treat semilinear evolution equations in Banach spaces. Convergence results are proved for variable step size implementations. To demonstrate the efficiency of the new integrators, we give some numerical experiments in MATLAB. In particular, numerical comparisons for semilinear parabolic PDEs in one and two space dimensions are presented.