Analysis of variable-step/non-autonomous artificial compression methods

Abstract

A standard artificial compression (AC) method for incompressible flow is $$\begin{gathered} \frac{u_{n+1}^{\epsilon }-u_n^{\epsilon }}{k}+u_{n+1}^{\epsilon }\cdot \nabla u_{n+1}^{\epsilon }+\frac{1}{2}{u}_{n+1}^{\epsilon }\nabla \cdot u_{n+1}^{\epsilon }+\nabla p_{n+1}^{\epsilon }-\nu \Delta u_{n+1}^{\epsilon }=f\text{ ,} \\ \epsilon \frac{p_{n+1}^{\epsilon }-p_n^{\epsilon }}{k}+\nabla \cdot u_{n+1}^{\epsilon }=0 \end{gathered}$$ for, typically, $\epsilon =k$ (timestep). It is fast, efficient and stable with accuracy $O(\epsilon +k)$. For adaptive (and thus variable) timestep $k_n$ (and thus $\epsilon ={\epsilon }_n$) its long time stability is unknown. For variable $k,\epsilon$ 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 $\epsilon =\epsilon (t)$\ artificial compression model to a weak solution of the incompressible Navier-Stokes equations as $\epsilon =\epsilon (t)\to 0$. The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variable $\epsilon ,k$ numerical tests in $2d$ and $3d$ are given for the new AC method.

22 pages