Conditions for $L^2$-dissipativity of an explicit symmetric finite-difference scheme for linearized 2d and 3d gas dynamics equations with a regularization

Abstract

<p style='text-indent:20px;'>We study an explicit two-level in time and symmetric in space finite-difference scheme for a linearized 2D and 3D gas dynamic system of equations with a kinetic-type regularization. For an initial-boundary value problem on any nonuniform rectangular mesh, sufficient Courant-type conditions for the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-dissipativity are derived for the first time by the energy method. For the Cauchy problem on the uniform mesh, recent both necessary conditions and sufficient conditions for the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-dissipativity in the spectral method are improved. A new form for the relaxation parameter is suggested which guarantees that the Courant-type number is uniformly bounded from above and below with respect to the mesh and the Mach number.</p>