  We consider a higher-order Milstein scheme for stochastic partial
differential equations with trace class noise which fulfill a certain
commutativity condition. A novel technique to generally improve the order of
convergence of Taylor schemes for stochastic partial differential equations 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 to directly compare various numerical schemes in terms
of their efficiency. As the main result, we prove for a broad class of
stochastic partial differential equations, 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.
