Magnus Exponential Integrators for Stiff Time-Varying Stochastic Systems

TL;DR

This paper introduces exponential numerical integrators for stiff, time-varying stochastic systems that preserve statistical structures and handle non-commuting operators, improving simulation accuracy for physical and spatially-extended systems.

Contribution

It develops novel Magnus expansion-based stochastic integrators for non-commuting, time-dependent operators, addressing a key challenge in simulating stiff stochastic systems.

Findings

Effective in particle simulations and SPDEs

Preserve fluctuation-dissipation balance

Handle non-commuting operators without direct stochastic integral evaluation

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.