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arXiv · 2504.13374

The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation

Andreas Wagner·Barbara Wohlmuth·Jan Zawallich
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Abstract

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 $t^{n+k}$ for $k\geq3$. 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 $H^1$-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.

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Andreas Wagner, Barbara Wohlmuth, Jan Zawallich. 2025-04-17. The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation. https://arxiv.org/abs/2504.13374

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