Abstract:
At first splitting integrators for linear systems of stochastic (Stratonovich-) differential equations with constant, non-commuting coefficients concerning the strong convergence are explored. By means of a stochastic Lady-Windermere fan and shuffle algebra structures a minimal system of order conditions for a given strong global order of such schemes is derived. Afterwards splitting methods of strong global order 1 and 1.5 are determined, and it is shown, that these methods converge with the corresponding order in the case of time-dependent and sufficiently smooth coefficients, too. Furthermore it is proven, that under the assumption of discrete commutator-bounds the stochastic (Strang-) Theta-Splitting applied to a pseudo-spectral discretized stochastic Schrödinger equation is of strong global order 1. Despite of the occurrence of an operator having arbitrarily large eigenvalues, these estimates are free of step-size restrictions and only need H^1-regularity of the exact solution. All convergence results are confirmed by numerical computer experiments.