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We study an explicit two-level symmetric (in space) finite-difference scheme for the multidimensional barotropic gas dynamics system of equations with quasi-gasdynamic regularization linearized at a constant solution (with an arbitrary velocity). A criterion and both necessary and sufficient conditions for the L2-dissipativity of the solutions to the Cauchy problem for the scheme are derived by the spectral method. In them, the Courant number is uniformly bounded with respect to the Mach number.

We study a symmetric three-level (in time) method with a weight and a symmetric vector two-level method for solving the initial-boundary value problem for a second-order hyperbolic equation with a small parameter $$\tau > 0$$ multiplying the highest time derivative, where the hyperbolic equation is a perturbation of the corresponding parabolic equation. It is proved that the solutions are uniformly stable in $$\tau $$ and time in two norms with respect to the initial data and the right-hand side of the equation. Additionally, the case where $$\tau $$ also multiplies the elliptic part of the equation is covered. The spacial discretization can be performed using the finite-difference or finite element method.

An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.

Entropy balance in the one-dimensional hyperbolic quasi-gasdynamic (HQGD) system of equations is analyzed. In particular, in regular flow regimes, it is shown that the behavior of entropy in the HQGD system is mainly determined by terms involving the natural viscosity and thermal conductivity coefficients. The total entropy production differs from the Navier–Stokes equations for viscous compressible heat-conducting gases by O(τ2) terms, where τ is a relaxation parameter. Additionally, a similar analysis of energy balance is performed for the simpler case of the barotropic HQGD system, which is of interest for some applications.

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.

The multidimensional quasi-gasdynamic system of equations written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization on a nonuniform rectangular grid is constructed for this system. The basic unknown functions (density, velocity, and temperature) are defined on a common grid, while the fluxes and viscous stresses, on staggered grids. The discretization is specially constructed so that the total entropy does not decrease, which is achieved by applying numerous original features.

The quasigasdynamic (QGD) approach makes it possible to construct convenient and reliable differ� ence schemes for the numerical solution of various gasdynamic problems. Its description can be found in (1-3). More specifically, in (2, Chapter 9) (see also (4)), the Boltzmann kinetic equation for a mixture of monatomic gases (5) is used to derive and test QGD equations for binary mixtures of nonreactive ideal polytropic gases. In this paper, we analyze and expand the capabili� ties of the QGD approach in this area. The original equations from (2) are rewritten as conservation laws, which are more conventional in viscous gas dynamics and convenient for discretization. Additionally, an external force and a heat source are taken into account. We briefly discuss the parabolicity of the sys� tem in the sense of Petrovskii, which ensures that the system is well defined. An entropy balance equation is derived, and the entropy production for a gas mixture is shown to be nonnegative, which ensures that the sys� tem is physically consistent (but does not hold in all available descriptions of gas mixtures). Importantly, to achieve the latter property, the expressions for the exchange terms in the total energy balance equation (initially derived only for monatomic gas mixtures) are properly generalized. Additionally, we introduce a simplification of the QGD system for binary mixtures, which is referred to as a quasihydrodynamic system and is used for the numerical simulation of weakly compressible suband transonic flows. At the end of this paper, we present simplified barotropic versions of both systems and derive a corresponding energy bal� ance equation with nonpositive energy production. The QGD system for binary gas mixtures a and b (see (2)) consists of the following mass balance, momentum, and total energy equations for the gas α

The classical Euler system of equations, which is mathematically a system of first-order quasilinear hyperbolic equations, describes inviscid gas flows. There are numerous modifications of the system, primarily including the viscous Navier‐Stokes equations, which are of the compound hyperbolic‐parabolic type. This paper addresses different modifications. First, we analyze a typical quasi-gasdynamic system of equations from [1]. It is shown that the system is parabolic in the sense of Petrovskii (in the physically most interesting range of parameters). Second, we consider the generalized Euler system of equations from [2]. It is established that the system is Petrovskii elliptic (which, in our view, makes its application problematic from the point of view of mechanics and computational mathematics). Third, two related modifications are examined, which are shown to be systems of secondorder quasilinear hyperbolic equations. Sketches of the proofs are presented. 1. The system of quasi-gasdynamic equations from