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.

翻译：我们证明了环面D上二维Bénard-Boussinesq模型的半隐式时间欧拉格式具有收敛性。对于乘性噪声，其依概率收敛速率接近1/2；这依赖于对过程及其格式在多种范数下的矩估计。在加性噪声情形下，由于方程组间的耦合作用，当环面顶部与底部之间的温差相对于粘性系数和热扩散系数不过大时，我们证明了速度场在$(L^2(D))^2$空间与温度场在$L^2(D)$空间中均具有强多项式收敛速率（接近1/2）。该结果依赖于格式的指数矩估计；由于两个演化方程均包含涉及另一物理量的线性项，证明必须同时对速度场和温度场进行。两种情形下的收敛速率均与Navier-Stokes方程所获结果相似。