Long-time stable SAV-BDF2 numerical schemes for the forced Navier-Stokes equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913538562326528 |
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| author | Han, Daozhi Wang, Xiaoming |
| author_facet | Han, Daozhi Wang, Xiaoming |
| contents | We propose a novel second-order accurate, long-time unconditionally stable time-marching scheme for the forced Navier-Stokes equations. A new Forced Scalar Auxiliary Variable approach (FSAV) is introduced to preserve the underlying dissipative structure of the forced system that yields a uniform-in-time estimate of the numerical solution. In addition, the numerical scheme is autonomous if the underlying model is, laying the foundation for studying long-time dynamics of the numerical solution via dynamical system approach. As an example we apply the new algorithm to the two-dimensional incompressible Navier-Stokes equations. In the case with no-penetration and free-slip boundary condition on a simply connected domain, we are also able to derive a uniform-in-time estimate of the vorticity in $H^1$ norm in addition to the $L^2$ norm guaranteed by the general framework. Numerical results demonstrate superior performance of the new algorithm in terms of accuracy, efficiency, stability and robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time stable SAV-BDF2 numerical schemes for the forced Navier-Stokes equations Han, Daozhi Wang, Xiaoming Numerical Analysis 35K61 76T99, 76S05, 76D07 We propose a novel second-order accurate, long-time unconditionally stable time-marching scheme for the forced Navier-Stokes equations. A new Forced Scalar Auxiliary Variable approach (FSAV) is introduced to preserve the underlying dissipative structure of the forced system that yields a uniform-in-time estimate of the numerical solution. In addition, the numerical scheme is autonomous if the underlying model is, laying the foundation for studying long-time dynamics of the numerical solution via dynamical system approach. As an example we apply the new algorithm to the two-dimensional incompressible Navier-Stokes equations. In the case with no-penetration and free-slip boundary condition on a simply connected domain, we are also able to derive a uniform-in-time estimate of the vorticity in $H^1$ norm in addition to the $L^2$ norm guaranteed by the general framework. Numerical results demonstrate superior performance of the new algorithm in terms of accuracy, efficiency, stability and robustness. |
| title | Long-time stable SAV-BDF2 numerical schemes for the forced Navier-Stokes equations |
| topic | Numerical Analysis 35K61 76T99, 76S05, 76D07 |
| url | https://arxiv.org/abs/2410.06362 |