Higher order time discretization method for the stochastic Stokes equations with multiplicative noise

TL;DR

This paper introduces a higher order time discretization method for stochastic Stokes equations with multiplicative noise, achieving strong convergence and providing new insights into pressure approximation and computational interpretation.

Contribution

A novel higher order time discretization approach based on the Milstein method for stochastic Stokes equations, with detailed error analysis and a new pressure interpretation.

Findings

Strong convergence order of at most 1 for velocity and pressure

Error estimates validated by numerical experiments

New interpretation of pressure solution for computational purposes

Abstract

In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of at most $1$ for both velocity and pressure approximations. The proof is based on a new H\"older continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard $L^2$- and $H^1$-norms, the pressure errors are carefully analyzed in a special norm because of the low regularity of the pressure solution. In addition, a new interpretation of the pressure solution, which is very useful in computation, is also introduced. Numerical experiments are also provided to validate the error estimates and their sharpness.