Analysis of Variable-Step/Non-autonomous Artificial Compression Methods
Analysis of Variable-Step/Non-autonomous Artificial Compression Methods
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DOI：
10.1007/s00021-019-0429-2
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发表时间：
2018-09
影响因子：
1.3
通讯作者：
R. Chen;W. Layton;Michael McLaughlin
中科院分区：
文献类型：
--
作者：
R. Chen;W. Layton;Michael McLaughlin
A standard artificial compression (AC) method for incompressible flow is $$\begin{aligned}&\frac{u_{n+1}^{\varepsilon }-u_{n}^{\varepsilon }}{k}+u_{n+1}^{\varepsilon }\cdot \nabla u_{n+1}^{\varepsilon }+{\frac{1}{2}}u_{n+1}^{\varepsilon }\nabla \cdot u_{n+1}^{\varepsilon }+\nabla p_{n+1}^{\varepsilon }-\nu \Delta u_{n+1}^{\varepsilon }=f\text { ,} \\&\quad \varepsilon \frac{p_{n+1}^{\varepsilon }-p_{n}^{\varepsilon }}{k} +\nabla \cdot u_{n+1}^{\varepsilon }=0 \end{aligned}$$for, typically,(timestep). It is fast, efficient and stable with accuracy. For adaptive (and thus variable) timestep(and thus) its long time stability is unknown. For variablethis report shows how to adapt a standard AC method to recover a provably stable method. For the associated continuum AC model, we prove convergence of theartificial compression model to a weak solution of the incompressible Navier–Stokes equations as. The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variablenumerical tests in 2dand 3dare given for the new AC method.
A standard artificial compression (AC) method for incompressible flow is $$\begin{aligned}&\frac{u_{n+1}^{\varepsilon }-u_{n}^{\varepsilon }}{k}+u_{n+1}^{\varepsilon }\cdot \nabla u_{n+1}^{\varepsilon }+{\frac{1}{2}}u_{n+1}^{\varepsilon }\nabla \cdot u_{n+1}^{\varepsilon }+\nabla p_{n+1}^{\varepsilon }-\nu \Delta u_{n+1}^{\varepsilon }=f\text { ,} \\&\quad \varepsilon \frac{p_{n+1}^{\varepsilon }-p_{n}^{\varepsilon }}{k} +\nabla \cdot u_{n+1}^{\varepsilon }=0 \end{aligned}$$for, typically,(timestep). It is fast, efficient and stable with accuracy. For adaptive (and thus variable) timestep(and thus) its long time stability is unknown. For variablethis report shows how to adapt a standard AC method to recover a provably stable method. For the associated continuum AC model, we prove convergence of theartificial compression model to a weak solution of the incompressible Navier–Stokes equations as. The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variablenumerical tests in 2dand 3dare given for the new AC method.