Invariant solutions of one-dimensional equations of two-temperature relaxation gas dynamics

The group analysis method is applied to the plane one-dimensional equations of two-temperature gas dynamics. The complete classification of the equations with respect to admitted Lie group is studied. All invariant solutions are analyzed and their comparisons with the known invariant solutions of the ideal gas dynamics system are presented.     

The non-stationary invariant solution of the equations of gas dynamics describing the spreading of a gas into a vacuum

An invariant solution of the equations of gas dynamics, constructed on a one-dimensional subgroup (according to the classification in /1/) which is only allowed in the case of a polytropic gas with a special adiabatic index, is considered. The gas spreads out into a vacuum after a finite time. New solutions are constructed which describe one-dimensional flows from a source into a vacuum and the focussing of the gas within a sphere or a cylinder with shock waves. The spreading of a concentration of the gas with an arbitrary boundary when there is a contact discontinuity is also considered.

One-dimensional flows have been treated in detail in /2, 3/, mainly in the case of extended subgroups.     

Some properties of difference schemes in one-dimensional gas dynamics

The linear implicit difference scheme is analyzed on specimen problems using numerical computations. Stability of some difference schemes is investigated, whose solvability is improved by introducing explicitness. A stability theorem is proved for the difference scheme of one-dimensional gas dynamics in the acoustic approximation near an arbitrary smooth solution.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 20–28, 1985.     

Unified solutions to some classical problems in rarefied gas dynamics based on the one-dimensional linearized S-model equations

An analytical version of the discrete-ordinates method is used to derive solutions for a class of problems defined in terms of the S-model kinetic equations. In addition to a general derivation, which is common to all the problems, specific analytical and computational aspects for each one of the problems are presented. In particular, numerical results for velocity profile, heat-flow profile and flow rates are obtained with high accuracy for the plane channel problems. In the case of half-space problems, the thermal and viscous-slip coefficients are also listed. Received: September 20, 2004; revised: March 31, 2005     

Solution of Kirkwood-Salsburg equations for a one-dimensional lattice gas

Expressions are obtained for the lowest correlation functions directly from the Kirkwood-Salsburg equations for an infinite system of particles on a one-dimensional lattice with two-body nearest-neighbor interaction in certain external fields. The problem of finding the external field that makes the density oscillation near the wall uniform is considered.State University, Tyumen'. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 463–469, March, 1995.     

On the Riemann matrices of the hyperbolic system dual to the system of equations of one-dimensional gas dynamics

We consider the Cauchy problem for the quasilinear hyperbolic system describing a one-dimensional flow of a gas with the equation of state p = p(ϱ), p′(ϱ) > 0, and with initial data satisfying a monotonicity condition. We suggest an approach to solving it by reduction to the Cauchy problem for the linear hyperbolic system obtained from the original system by the hodograph transformation. These constructions are extended to a system of elasticity equations describing nonlinear vibrations of a one-dimensional medium. The main result is illustrated by two examples.     

Integral equations for correlation functions of a quantum one-dimensional Bose gas

The large-time, long-distance behavior of the temperature correlation functions of a quantum one-dimensional Bose gas is considered. We obtain integral equations, which are closely related to the thermodynamic Bethe ansatz equations and whose solutions describe asymptotic expressions. In the low-temperature limit, the solutions of these equations are expressed through observables of the model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 117–138, October, 1999.     

About the equations of non-stationary magneto-hydrodynamics

We obtain weak solutions for the equations of magneto-hydrodynamics which are regular for almost all times. If these solutions are small in a suitable sense they are strong for all times. The conductivity, and to a lesser extent the magnetic permeability, are allowed to vary discontinuously.     

A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

Symmetrization of the nonlinear system of gas dynamics equations

We describe a method for representing the nonlinear system of gas dynamics equations in quasilinear form with symmetric coefficient matrices and, moreover, with a positive definite matrix at the time derivative.     

Quasi-acoustic scheme for the Euler equations of gas dynamics

We suggest an original scheme and an algorithm for the numerical solution of the Euler equations of gas dynamics. The construction of the scheme is based on the mass, momentum, and energy conservation laws. The flux computation is carried out by summation of elementary fluxes formed by small-amplitude running waves that satisfy the linearized equations of gas dynamics. The scheme contains no artificial regularizers, has second-order accuracy on smooth solutions, and is quasimonotone in a neighborhood of the discontinuities. Examples of one- and two-dimensional computations are given.