The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916695959928832 |
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| author | Wagner, Andreas Wohlmuth, Barbara Zawallich, Jan |
| author_facet | Wagner, Andreas Wohlmuth, Barbara Zawallich, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation Wagner, Andreas Wohlmuth, Barbara Zawallich, Jan Numerical Analysis 46N40, 65M12, 65M15 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. |
| title | The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation |
| topic | Numerical Analysis 46N40, 65M12, 65M15 |
| url | https://arxiv.org/abs/2504.13374 |