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Rate of convergence of a semi-implicit time Euler scheme for a 2D Bénard-Boussinesq model

Hakima Bessaih, Annie Millet

TL;DR

This work analyzes a semi-implicit time Euler discretization for the 2D stochastic Bénard-Boussinesq system on the torus, addressing both multiplicative and additive noise. Using moment and exponential-moment estimates, localized convergence, and careful coupling of the velocity and temperature equations, it establishes an almost $\tfrac{1}{2}$ rate in probability for multiplicative noise and a strong near-$\tfrac{1}{2}$-order convergence (up to logarithmic factors) for additive noise under a small temperature-difference constraint $|C_L|$. The results extend prior BeMi_Bou analyses to the Bénard setting, proving convergence, moment bounds, and exponential-moment control for the semi-implicit scheme, and quantify how viscosity $\nu$, thermal diffusivity $\kappa$, and noise covariance interact to determine rates. The findings provide rigorous guidance for the numerical approximation of stochastic Boussinesq flows, including explicit conditions under which strong convergence is guaranteed and rates are optimized. Appendix material supports the main results with detailed proofs and extensions.

Abstract

We prove that a semi-implicit time Euler scheme for the two-dimensional Bénard-Boussinesq model on the torus D converges. The rate of convergence in probability is almost 1/2 for a multiplicative noise; this relies on moment estimates in various norms for the processes and the scheme. In case of an additive noise, due to the coupling of the equations, provided that the difference on temperature between the top and bottom parts of the torus is not too big compared to the viscosity and thermal diffusivity, a strong polynomial rate of convergence (almost 1/2) is proven in $(L^2(D))^2$ for the velocity and in $L^2(D)$ for the temperature. It depends on exponential moments of the scheme; due to linear terms involving the other quantity in both evolution equations, the proof has to be done simultaneaously for both the velocity and the temperature. These rates in both cases are similar to that obtained for the Navier-Stokes equation.

Rate of convergence of a semi-implicit time Euler scheme for a 2D Bénard-Boussinesq model

TL;DR

This work analyzes a semi-implicit time Euler discretization for the 2D stochastic Bénard-Boussinesq system on the torus, addressing both multiplicative and additive noise. Using moment and exponential-moment estimates, localized convergence, and careful coupling of the velocity and temperature equations, it establishes an almost rate in probability for multiplicative noise and a strong near--order convergence (up to logarithmic factors) for additive noise under a small temperature-difference constraint . The results extend prior BeMi_Bou analyses to the Bénard setting, proving convergence, moment bounds, and exponential-moment control for the semi-implicit scheme, and quantify how viscosity , thermal diffusivity , and noise covariance interact to determine rates. The findings provide rigorous guidance for the numerical approximation of stochastic Boussinesq flows, including explicit conditions under which strong convergence is guaranteed and rates are optimized. Appendix material supports the main results with detailed proofs and extensions.

Abstract

We prove that a semi-implicit time Euler scheme for the two-dimensional Bénard-Boussinesq model on the torus D converges. The rate of convergence in probability is almost 1/2 for a multiplicative noise; this relies on moment estimates in various norms for the processes and the scheme. In case of an additive noise, due to the coupling of the equations, provided that the difference on temperature between the top and bottom parts of the torus is not too big compared to the viscosity and thermal diffusivity, a strong polynomial rate of convergence (almost 1/2) is proven in for the velocity and in for the temperature. It depends on exponential moments of the scheme; due to linear terms involving the other quantity in both evolution equations, the proof has to be done simultaneaously for both the velocity and the temperature. These rates in both cases are similar to that obtained for the Navier-Stokes equation.

Paper Structure

This paper contains 14 sections, 17 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Description of the model
  3. The functional framework
  4. The multiplicative stochastic perturbation
  5. Global well posedness and moment estimates of $(u,\theta)$
  6. The semi-implicit time Euler scheme with multiplicative noise
  7. Moments of the semi-implicit time Euler scheme
  8. Strong convergence of a localized semi-implicit time Euler scheme
  9. Rate of Convergence in Probability and in $L^2(\Omega)$
  10. Rate of Convergence in Probability
  11. Rate of Convergence in $L^2(\Omega)$
  12. The additive stochastic perturbation - Exponential moments
  13. Strong convergence of the semi-implicit time Euler scheme - additive noise
  14. Appendix

Key Result

Lemma 2.1

Let $\alpha, \rho$ be positive numbers and $\delta \in [0,1)$ be such that $\delta +\rho > \frac{1}{2}$ and $\alpha + \delta +\rho \geq 1$. Let $u\in V^\alpha$, $v\in V^\rho$ and $\theta \in H^\rho$; then for some positive constant $C:= C(\alpha, \delta, \rho)$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • ...and 17 more