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<p>This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion around <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t^{n+k}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. An exponential time integrator is used for the auxiliary variable to ensure stability independent of the time step size. We give rigorous asymptotic error estimates of the time-stepping scheme, thereby justifying its accuracy and stability. The scheme is reformulated into one amenable to a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-conforming finite element discretization. Finally, we validate our theoretical results with numerical experiments using a Taylor–Hood-based finite element discretization and show its applicability to large-scale 3-dimensional problems.</p>

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The Generalized Scalar Auxiliary Variable Applied to the Incompressible Boussinesq Equation

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  • 出版模式：

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  • 发表日期：

    2026年3月24日

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  • Andreas Wagner,
  • Barbara Wohlmuth,
  • Jan Zawallich

摘要

This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion around \(t^{n+k}\) t n + k for \(k\ge 3\) k ≥ 3 . An exponential time integrator is used for the auxiliary variable to ensure stability independent of the time step size. We give rigorous asymptotic error estimates of the time-stepping scheme, thereby justifying its accuracy and stability. The scheme is reformulated into one amenable to a \(H^1\) H 1 -conforming finite element discretization. Finally, we validate our theoretical results with numerical experiments using a Taylor–Hood-based finite element discretization and show its applicability to large-scale 3-dimensional problems.