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Matem. Mod., 2021, Volume 33, Number 5, Pages 16–34 (Mi mm4284)  

$L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small Mach numbers

A. A. Zlotnikab, T. A. Lomonosova

a NRU Higher School of Economics
b Keldysh Institute of Applied Mathematics of RAS

Abstract: We study explicit two-level finite-difference schemes on staggered meshes for two known regularizations of $\mathrm{1D}$ barotropic gas dynamics equations including schemes with discretizations in $x$ that possess the dissipativity property with respect to the total energy. We derive criterions of $L^2$-dissipativity in the Cauchy problem for their linearizations at a constant solution with zero background velocity. We compare the criterions for schemes on non-staggered and staggered meshes. Also we consider the case of $\mathrm{1D}$ Navier–Stokes equations without artificial viscosity coefficient. For one of their regularizations, the maximal time step is guaranteed for the choice of the regularization parameter $\tau_{opt}=\nu_*/c^2_*$, where $c_*$ and $\nu_*$ are the background sound speed and kinematic viscosity; such a choice does not depend on the meshes. To analyze the case of the $\mathrm{1D}$ Navier–Stokes–Cahn–Hilliard equations, we derive and verify the criterions for $L^2$-dissipativity and stability for an explicit finite-difference scheme approximating a nonstationary $4^{th}$-order in $x$ equation that includes a $2^{nd}$-order term in $x$. The obtained criteria may be useful to compute flows at small Mach numbers.

Keywords: $L^2$-dissipativity, explicit finite-difference schemes, staggered meshes, gas dynamics equations, Navier–Stokes–Cahn–Hilliard equations.

Funding Agency Grant Number
Russian Science Foundation 19-11-00169


DOI: https://doi.org/10.20948/mm-2021-05-02

Full text: PDF file (506 kB)
First page: PDF file
References: PDF file   HTML file

Received: 11.02.2021
Revised: 11.02.2021
Accepted:15.03.2021

Citation: A. A. Zlotnik, T. A. Lomonosov, “$L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small Mach numbers”, Matem. Mod., 33:5 (2021), 16–34

Citation in format AMSBIB
\Bibitem{ZloLom21}
\by A.~A.~Zlotnik, T.~A.~Lomonosov
\paper $L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small Mach numbers
\jour Matem. Mod.
\yr 2021
\vol 33
\issue 5
\pages 16--34
\mathnet{http://mi.mathnet.ru/mm4284}
\crossref{https://doi.org/10.20948/mm-2021-05-02}


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