Abstract
A standard artificial compression (AC) method for incompressible flow is
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.
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R. M. Chen: The research herein was partially supported by NSF Grant DMS 1613375. W. Layton: The research herein was partially supported by NSF Grants DMS1522267, 1817542 and CBET 1609120. M. McLaughlin: The research herein was partially supported by NSF Grants DMS1522267 and 1817542.
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Chen, R.M., Layton, W. & McLaughlin, M. Analysis of Variable-Step/Non-autonomous Artificial Compression Methods. J. Math. Fluid Mech. 21, 30 (2019). https://doi.org/10.1007/s00021-019-0429-2
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DOI: https://doi.org/10.1007/s00021-019-0429-2