A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case

Abstract We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need fulfill commutativity condition for noise term. The newly developed differs significantly from schemes are appropriate case commutative noise. As key result, new needs only two stages specifically tailored based on technique that, compared original scheme, allows reduction computational complexity by one order magnitude. Moreover, proposed can flexibly be combined with some approximation method involved iterated integrals. main we prove strong $$L^2$$ L2 -convergence introduced especially if it is any suitable algorithm necessary carry out rigorous analysis error versus cost and derive effective convergence in truncated Fourier series integrals applied. further novelty, show use approximations together improves Euler scheme. This result contrary well known results finite dimensional SDE where merely does improve -sense