An Efficient Derivative-Free Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise

We propose a derivative-free Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. The same theoretical order of convergence with respect to the spatial and time discretizations as for the Milstein scheme is obtained whereas the computational cost is, in general, considerably lower. As the main result, we show that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than the effective order of convergence of the original Milstein scheme if errors versus computational costs are considered. In addition, the derivative-free Milstein scheme is efficiently applicable to general semilinear stochastic partial differential equations which do not have to be multiplicative in the $Q$-Wiener process. Finally we prove the convergence of the proposed scheme. This version of the paper presents work in progress.