Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model

Hakima Bessaih (1) , Annie Millet (2, 3)

1 FIU - Florida International University [Miami]
2 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We prove that an implicit time Euler scheme for the 2D-Boussinesq model on the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity and temperature, as well as their discretizations, are computed. We obtain the optimal speed of convergence in probability, and a logarithmic speed of convergence in $L^2(\Omega)$. These results are deduced from a time regularity of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(\Omega)$ convergence restricted to a subset where the $W^{1,2}$-noms of the solutions are bounded.

Mots clés: Boussinesq model; implicit time Euler scheme; convergence in probability; strong convergence

Domaines: Probabilités [math.PR]

Hakima Bessaih, Annie Millet. Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model. Mathematics , 2022, Computational Methods in Nonlinear Analysis and Their Applications, 10 (22), pp.4246. ⟨10.3390/math10224246⟩. ⟨hal-03812476⟩