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We consider a higher-order Milstein scheme for stochastic partialdifferential equations with trace class noise which fulfill a certaincommutativity condition. A novel technique to generally improve the order ofconvergence of Taylor schemes for stochastic partial differential equations isintroduced. The key tool is an efficient approximation of the Milstein term byparticularly tailored nested derivative-free terms. For the resultingderivative-free Milstein scheme the computational cost is, in general,considerably reduced by some power. Further, a rigorous computational costmodel is considered and the so called effective order of convergence isintroduced which allows to directly compare various numerical schemes in termsof their efficiency. As the main result, we prove for a broad class ofstochastic partial differential equations, including equations with operatorsthat do not need to be pointwise multiplicative, that the effective order ofconvergence of the proposed derivative-free Milstein scheme is significantlyhigher than for the original Milstein scheme. In this case, the derivative-freeMilstein scheme outperforms the Euler scheme as well as the original Milsteinscheme due to the reduction of the computational cost. Finally, we present somenumerical examples that confirm the theoretical results.

Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise