A Higher Order Discretization for the Stochastic Navier--Stokes equations with additive Noise

TL;DR

The paper develops a higher-order time discretization for the stochastic Navier–Stokes equations with additive noise by reformulating the system as a random PDE through the transform $y = u - \Phi W$. A Modified Crank–Nicolson scheme is designed with Brownian-quadrature corrections to achieve a strong convergence rate of $O(\tau^{3/2})$ in probability for both velocity and pressure, and the analysis includes detailed residual bounds, a localized Grönwall argument, and an inf–sup pressure estimate. The results extend to general noise via Helmholtz decomposition and include a linear Stokes benchmark, with numerical experiments on academic and lid-driven cavity problems confirming the theoretical rates and demonstrating practical efficiency. The work provides a robust framework for accurate, higher-order stochastic simulations of incompressible flows with additive noise in 2D, with potential extensions to more complex domains and non-Newtonian contexts.

Abstract

We propose a new higher-order time discretization scheme for the stochastic Navier--Stokes equations with additive noise, where its velocity and pressure approximates converge at strong rate $1.5$ in probability. The construction rests on its reformulation as a random PDE for the transform $y = u- ΦW$, and different higher order numerical quadrature rules for the diffusion and the drift part. The theoretical findings are supported by numerical simulations.

Figures (7)

• Figure 1: Lid-driven cavity (see Section \ref{['example_lid_driven']}): Streamlines of the deterministic solution ($\mu = 0$; left), streamlines of the expected value of the solution with smaller ($\mu=10$; middle), and larger noise ($\mu=40$; right) at $T=20$.
• Figure 2: Academic Example in Section \ref{['academic_example']}: approximation error of the CN scheme for the velocity and the pressure for Navier--Stokes equations (left) and Stokes equations (right).
• Figure 3: Error of the approximation of the velocity (left) and the pressure (right).
• Figure 4: Streamlines of the deterministic solution of the CN scheme ($\mu=0$) at $T=100$ (left), streamlines of the time-averaged solution of the CN scheme with $\mu=10$ (middle) and $\mu=40$ (right).
• Figure 5: Solution of the CN scheme for one realisation of the noise with $\mu=10$ at time $t=10,20,50$.

Theorems & Definitions (30)

• Lemma 2.1: Transport-form energy cancellation
• Definition 2.1
• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.1
• proof
• Lemma 2.3
• proof
• Theorem 3.1: First main result