Numerical methods for stochastic partial differential equations with multiples scales

TL;DR

This paper introduces a novel numerical approach combining spectral and multiscale methods to efficiently solve stochastic PDEs with multiple scales, accurately capturing long-term dynamics without high computational costs.

Contribution

The paper presents a new hybrid spectral-multiscale method for SPDEs with quadratic nonlinearities, enabling efficient long-term simulation independent of small-scale complexities.

Findings

Accurately captures amplitude equations at long time scales.

Cost remains independent of small-scale features.

Numerical experiments validate the method's effectiveness.

Abstract

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.