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<p>We prove that a semi-implicit time Euler scheme for the two-dimensional Bénard–Boussinesq model on the torus <i>D</i> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((L^2(D))^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> for the velocity and in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2(D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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Rate of convergence of a semi-implicit time euler scheme for a 2D bénard–boussinesq model

  • 期刊论文

  • 发表日期：

    2025年1月31日

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  • Hakima Bessaih,
  • Annie Millet

摘要

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 \((L^2(D))^2\) ( L 2 ( D ) ) 2 for the velocity and in \(L^2(D)\) 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.