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

• Robin Ming Chen
• William Layton
• 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

### Conflict of interest

The authors declare that they have no conflicts of interest.

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## Authors and Affiliations

• Robin Ming Chen
• 1
• William Layton
• 1
• Michael McLaughlin
• 1
1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA