Uniformly accurate schemes for drift--oscillatory stochastic differential equations

TL;DR

This paper develops uniformly accurate numerical schemes for drift-oscillatory stochastic differential equations, maintaining convergence orders across various oscillation regimes using micro-macro and integral methods.

Contribution

It introduces a micro-macro decomposition approach that preserves convergence orders and can be extended to higher order methods for oscillatory SDEs.

Findings

Micro-macro scheme achieves uniform weak order one and strong order one half.

Integral scheme also attains the same uniform accuracy orders.

Micro-macro scheme can be generalized to higher order methods.

Abstract

In this work, we adapt the {\em micro-macro} methodology to stochastic differential equations for the purpose of numerically solving oscillatory evolution equations. The models we consider are addressed in a wide spectrum of regimes where oscillations may be slow or fast. We show that through an ad-hoc transformation (the micro-macro decomposition), it is possible to retain the usual orders of convergence of Euler-Maruyama method, that is to say, uniform weak order one and uniform strong order one half. We also show that the same orders of uniform accuracy can be achieved by a simple integral scheme. The advantage of the micro-macro scheme is that, in contrast to the integral scheme, it can be generalized to higher order methods.