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Mean Square Temporal error estimates for the 2D stochastic Navier-Stokes equations with transport noise

Dominic Breit, Thamsanqa Castern Moyo, Andreas Prohl, Jörn Wichmann

TL;DR

An error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2 holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known withrespect to convergence in probability.

Abstract

We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.

Mean Square Temporal error estimates for the 2D stochastic Navier-Stokes equations with transport noise

TL;DR

An error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2 holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known withrespect to convergence in probability.

Abstract

We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.
Paper Structure (31 sections, 4 theorems, 115 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 31 sections, 4 theorems, 115 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Tensor calculus
  4. Function spaces
  5. Probability setup
  6. The concept of solutions
  7. Pressure decomposition
  8. Time-discretization
  9. Preparations
  10. Temporal error estimate
  11. Numerical simulations
  12. Fully discrete algorithm
  13. Finite Element Method
  14. Recovering the energy identity
  15. The algorithm
  16. ...and 16 more sections

Key Result

theorem 1

Let $(\Omega,\mathfrak{F},(\mathfrak{F}_t)_{t\geq0},\mathbb{P})$ be a given stochastic basis with a complete right-continuous filtration, $(W_k)_{k=1}^K$ being mutually independent real-valued standard Wiener processes relative to $(\mathfrak F_t)$, and $\bfsigma_k\in\mathbb R^2$, $k=1,\dots,K$. Let

Figures (5)

  • Figure 1: Graphical illustration of the results of SOE--1: time evolution of the kinetic energy with additive (), multiplicative (), transport () and no () noise. Thick lines and dotted lines show the mean energy and the mean energy plus or minus one standard deviation, respectively. The first 1,000 (out of 10,000) energy trajectories are shown in pale colours. Details on the numerical simulation are given in Section \ref{['sec:numerical-experiments']}.
  • Figure 2: Graphical illustration of the results of SOE--1: empirical approximation (based on 1,000 trajectories) of the stationary distributions of the kinetic energy for additive (), multiplicative () and transport () noise. The deterministic stationary energy level is indicated by a black () vertical line. Details on the numerical simulation are given in Section \ref{['sec:numerical-experiments']}.
  • Figure 3: Triangulation of the unit square that is used in all experiments.
  • Figure 4: Graphical illustration of the results of SOE--1: (left) time evolution of the kinetic energy for different noises. The deterministic kinetic energy evolution is shown in black. Thick lines and dotted lines show the mean energy and the mean energy plus or minus one standard deviation, respectively. The first 1,000 (out of 10,000) energy trajectories are shown in pale colours; (right) empirical approximation (based on 1,000 trajectories) of the stationary distributions of the kinetic energy for different noises. The deterministic stationary energy level is indicated by a black vertical line.
  • Figure 5: Graphical illustration of the results of SOE--2: time convergence of velocity and pressure. Colour encodes the experiments (Exp. 1: , Exp. 2: , Exp. 3: , Exp. 4: , Exp. 5: ). The discontinuous lines are reference lines where the individual slopes are given in each figures' legend.

Theorems & Definitions (6)

  • definition 1
  • theorem 1
  • corollary 1
  • corollary 2
  • theorem 2
  • proof