Rate of convergence of a semi-implicit time Euler scheme for a 2D B\'enard-Boussinesq model

TL;DR

This paper proves the convergence rate of a semi-implicit Euler scheme for the 2D Bénard-Boussinesq model, showing near 1/2 rate under certain noise conditions, with results comparable to Navier-Stokes equations.

Contribution

It establishes the convergence rate of a semi-implicit Euler scheme for the 2D Bénard-Boussinesq model, including cases with multiplicative and additive noise, under specific conditions.

Findings

Convergence rate is almost 1/2 for multiplicative noise.

Strong polynomial convergence rate is achieved for additive noise.

Rates are similar to those for the Navier-Stokes equations.

Abstract

We prove that a semi-implicit time Euler scheme for the two-dimensional B\'enard-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.