Conditions for dissipativity of an explicit finite-difference scheme for a linearized multidimensional quasi-gasdynamic system of equations

We study an explicit two-level finite-difference scheme for a linearized multidimensional quasi-gasdynamic system of equations. For an initial-boundary value problem on a nonuniform rectangular mesh, sufficient conditions of Courant-type for $L^2$-dissipativity are derived for the first time by applying the energy method. For the Cauchy problem on a uniform mesh, both necessary and sufficient conditions for $L^2$-dissipativity in the spectral method are improved. A new form of specifying the relaxation parameter is indicated which guarantees that the Courant-type number is uniformly bounded from above and below with respect to both the mesh and the Mach number.