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An efficient fully explicit scheme for stochastic Navier-Stokes equations driven by multiplicative noise

Can Huang, Weiwen Wang, Chuanju Xu

TL;DR

The paper develops a linear, fully explicit pressure-correction scheme for the 2D stochastic Navier–Stokes equations with multiplicative noise and Dirichlet boundaries by introducing two auxiliary variables that decouple the nonlinear convection and stochastic terms. The method remains unconditionally stable in the $2m$-th moment for all $m\\ge 1$ and reduces each time step to solving only Poisson-type problems with constant coefficients plus a small linear system, offering substantial computational efficiency. A strong convergence analysis is provided for the linearized equation, and numerical experiments corroborate the theoretical stability and convergence while demonstrating the method’s robustness under multiplicative noise. The approach extends explicit, stable discretizations to non-periodic domains and highlights the practical impact of auxiliary-variable techniques in stochastic flow simulations.

Abstract

This work proposes an efficient, linear, and fully decoupled pressure-correction scheme for the 2D stochastic Navier-Stokes equations with multiplicative noise and Dirichlet boundary condition. Leveraging the auxiliary variable approach, the scheme is fully explicit yet unconditionally stable. At each time step, it only requires solving Poisson-type equations with constant coefficients. To the best of our knowledge, this is the first application of the auxiliary variable method to stochastic Navier-Stokes equations. We provide a detailed strong convergence analysis for the linearized equation under standard assumptions.

An efficient fully explicit scheme for stochastic Navier-Stokes equations driven by multiplicative noise

TL;DR

The paper develops a linear, fully explicit pressure-correction scheme for the 2D stochastic Navier–Stokes equations with multiplicative noise and Dirichlet boundaries by introducing two auxiliary variables that decouple the nonlinear convection and stochastic terms. The method remains unconditionally stable in the -th moment for all and reduces each time step to solving only Poisson-type problems with constant coefficients plus a small linear system, offering substantial computational efficiency. A strong convergence analysis is provided for the linearized equation, and numerical experiments corroborate the theoretical stability and convergence while demonstrating the method’s robustness under multiplicative noise. The approach extends explicit, stable discretizations to non-periodic domains and highlights the practical impact of auxiliary-variable techniques in stochastic flow simulations.

Abstract

This work proposes an efficient, linear, and fully decoupled pressure-correction scheme for the 2D stochastic Navier-Stokes equations with multiplicative noise and Dirichlet boundary condition. Leveraging the auxiliary variable approach, the scheme is fully explicit yet unconditionally stable. At each time step, it only requires solving Poisson-type equations with constant coefficients. To the best of our knowledge, this is the first application of the auxiliary variable method to stochastic Navier-Stokes equations. We provide a detailed strong convergence analysis for the linearized equation under standard assumptions.

Paper Structure

This paper contains 15 sections, 106 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prerequisite results and Assumptions
  3. Notations and spaces
  4. Q-Wiener process
  5. Essential assumptions
  6. Exponential integrability of velocity
  7. Semi-discretization and its stability result
  8. Algorithm development
  9. Unconditional stability
  10. Why two auxiliary variables
  11. Error analysis for linearized equation
  12. Numerical experiments
  13. Accuracy test on homogeneous Dirichlet boundary condition
  14. Stochastic shear layer roll-up problem
  15. Concluding remarks

Figures (4)

  • Figure 3.1: Histogram produced by OAV method (KL divergence=$0.0190$) and TAV method (KL divergence=$0.0027$) for $\tau=T/700$.
  • Figure 5.1: Distribution of $\xi$ at $T$ with $g({\boldsymbol{u}})=1$ for various timesteps.
  • Figure 5.2: Distribution of $\eta$ at $T$ with $g({\boldsymbol{u}})=1$ for various timesteps.
  • Figure 5.3: Time evolution of vorticity contours at $t=0, 0.4, 0.8,1.2$. The top row: deterministic case. The bottom row: stochastic case with $g({\boldsymbol{u}})={ 0.1}$.

Theorems & Definitions (5)

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