Magnus Exponential Integrators for Stiff Time-Varying Stochastic Systems

Abstract: We introduce exponential numerical integration methods for stiff stochastic dynamical systems of the form $d\mathbf{z}_t = L(t)\mathbf{z}_tdt + \mathbf{f}(t)dt + Q(t)d\mathbf{W}_t$. We consider the setting of time-varying operators $L(t), Q(t)$ where they may not commute $L(t_1)L(t_2) \neq L(t_2)L(t_1)$, raising challenges for exponentiation. We develop stochastic numerical integration methods using Mangus expansions for preserving statistical structures and for maintaining fluctuation-dissipation balance for physical systems. For computing the contributions of the fluctuation terms, our methods provide alternative approaches without needing directly to evaluate stochastic integrals. We present results for our methods for a class of SDEs arising in particle simulations and for SPDEs for fluctuations of concentration fields in spatially-extended systems. For time-varying stochastic dynamical systems, our introduced discretization approaches provide general exponential numerical integrators for preserving statistical structures while handling stiffness.