Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise

Gilles Vilmart

Abstract. We introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise. We prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations, which permits to use large time steps. The approach is based on the micro-macro framework of multi-revolution composition methods recently introduced for deterministic problems and inherits its geometric features, in particular to design integrators preserving exactly quadratic first integral. Numerical experiments, including the stochastic nonlinear Schr╫dinger equation with space-time multiplicative noise, illustrate the performance and versatility of the approach.

Key Words. highly-oscillatory stochastic differential equation, composition method, quadratic first integral conservation, multiplicative noise, time-dependent stochastic Schr╫dinger equation.