Strong Stability Preservation for Stochastic Partial Differential Equations

TL;DR

This paper extends the concept of Strong Stability Preservation from deterministic to stochastic partial differential equations, ensuring stable numerical solutions that remain bounded, which benefits data assimilation involving SPDEs with monotonicity properties.

Contribution

It introduces a stochastic SSP framework that guarantees nonlinearly stable solutions for SPDEs, a novel extension of deterministic stability concepts.

Findings

Provides a stochastic SSP method for SPDEs

Ensures unconditionally bounded solutions in stochastic simulations

Potential applications in data assimilation with SPDEs

Abstract

This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) with pathwise solutions that remain unconditionally bounded. This approach may offer modelling advantages in data assimilation, particularly when the signal or data is a realization of an SPDE or PDE with a monotonicity property.