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Analysis of Variable-Step/Non-autonomous Artificial Compression Methods

  • Robin Ming Chen
  • William LaytonEmail author
  • Michael McLaughlin
Article

Abstract

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, \(\varepsilon =k\) (timestep). It is fast, efficient and stable with accuracy \(O(\varepsilon +k)\). For adaptive (and thus variable) timestep \(k_{n}\) (and thus \(\varepsilon =\varepsilon _{n}\)) its long time stability is unknown. For variable \(k,\varepsilon \) this 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 the \(\varepsilon =\varepsilon (t)\) artificial compression model to a weak solution of the incompressible Navier–Stokes equations as \(\varepsilon =\varepsilon (t)\rightarrow 0\). The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variable \(\varepsilon ,k\) numerical tests in 2d and 3d are given for the new AC method.

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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