Enhancing the order of the Milstein scheme for stochastic partial differential equations with commutative noise

We consider a higher-order Milstein scheme for stochastic partial differential equations (SPDEs) with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for SPDEs is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered, and the so-called effective order of convergence is introduced, which allows direct comparison of various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of SPDEs, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results.