Geodesic
$L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small Mach numbers
Matematičeskoe modelirovanie, Tome 33 (2021) no. 5, pp. 16-34 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $4^{\text{th}}$-order in $x$ equation that includes a $2^{\text{nd}}$-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
Mots-clés : Navier–Stokes–Cahn–Hilliard equations. 
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     author = {A. A. Zlotnik and T. A. Lomonosov},
     title = {$L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small {Mach} numbers},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {16--34},
     year = {2021},
     volume = {33},
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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. Matematičeskoe modelirovanie, Tome 33 (2021) no. 5, pp. 16-34. http://geodesic.mathdoc.fr/item/MM_2021_33_5_a1/

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