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Speed of Convergence of Time Euler Schemes for a Stochastic 2D Boussinesq Model

Author

Listed:
  • Hakima Bessaih

    (Mathematics and Statistics Department, Florida International University, 11200 SW 8th Street, Miami, FL 33199, USA)

  • Annie Millet

    (Statistique, Analyse et Modélisation Multidisciplinaire, EA 4543, Université Paris 1 Panthéon Sorbonne, Centre Pierre Mendès France, 90 Rue de Tolbiac, CEDEX, 75634 Paris, France
    Laboratoire de Probabilités, Statistique et Modélisation, UMR 8001, Universités Paris 6-Paris 7, Place Aurélie Nemours, 75013 Paris, France)

Abstract

We prove that an implicit time Euler scheme for the 2D Boussinesq model on the torus D converges. The various moments of the W 1 , 2 -norms of the velocity and temperature, as well as their discretizations, were computed. We obtained the optimal speed of convergence in probability, and a logarithmic speed of convergence in L 2 ( Ω ) . These results were deduced from a time regularity of the solution both in L 2 ( D ) and W 1 , 2 ( D ) , and from an L 2 ( Ω ) convergence restricted to a subset where the W 1 , 2 -norms of the solutions are bounded.

Suggested Citation

  • Hakima Bessaih & Annie Millet, 2022. "Speed of Convergence of Time Euler Schemes for a Stochastic 2D Boussinesq Model," Mathematics, MDPI, vol. 10(22), pages 1-39, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4246-:d:971322
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    References listed on IDEAS

    as
    1. Duan, Jinqiao & Millet, Annie, 2009. "Large deviations for the Boussinesq equations under random influences," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 2052-2081, June.
    2. Hannelore Breckner, 2000. "Galerkin approximation and the strong solution of the Navier-Stokes equation," International Journal of Stochastic Analysis, Hindawi, vol. 13, pages 1-21, January.
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