Implicit-explicit Crank-Nicolson scheme for Oseen's equation at high Reynolds number
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arXiv
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| Natura: | Preprint |
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2024
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| author | Burman, Erik Garg, Deepika Guzman, Johnny |
| author_facet | Burman, Erik Garg, Deepika Guzman, Johnny |
| contents | In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzmàn, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order $O(h^{k+\frac12} + τ^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $τ$ the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Implicit-explicit Crank-Nicolson scheme for Oseen's equation at high Reynolds number Burman, Erik Garg, Deepika Guzman, Johnny Numerical Analysis In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzmàn, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order $O(h^{k+\frac12} + τ^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $τ$ the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations. |
| title | Implicit-explicit Crank-Nicolson scheme for Oseen's equation at high Reynolds number |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2405.12562 |