Numerical Solution of the Godunov - Sultangazin System of Equations. Periodic Case

Abstract

The Cauchy problem of the Godunov - Sultangazin system of equations with periodic initial conditions is considered in the article. The Godunov - Sultangazin system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of three speeds. So, there are three groups of molecules. The molecules of the first two groups have the speeds equal in values and opposite in directions. The molecules of the third group have zero speed. The considered mathematical model has a number of properties of Boltzmann equation. This system of the equations is a quasi-linear system of partial differential equations. There is no analytic solution for this problem in the general case. So, numerical investigation of the Cauchy problem of the Godunov - Sultangazin system is very important. The finite-difference method of the first order is used for numerical investigation of the Cauchy problem of the Godunov - Sultangazin system of equations. The paper presents and discusses the results of numerical investigation of the Cauchy problem for the studied system solution with periodic initial condition. The dependence of the time of stabilization of the Cauchy problem solution of Godunov - Sultangazin system of equations from the decreasing parameter of system are obtained. The paper presents the dependence of time of energy exchange from the decreasing parameter. The solution stabilization to the equilibrium state is obtained. The stabilization time of Godunov - Sultangazin system solution is compared to the stabilization time of Carleman system solution in periodic case. The results of numerical investigation are in good agreement with the theoretical results obtained previously.

Secular Terms for the Kinetic Mckean Model

“…In what follows, we will consider the system (16) without the perturbation operator. Substituting (18) to (17) and taking into account that…”

“…The asymptotic stability of kinetic systems of Carleman, Godunov-Sultangazin and Broadwell for periodic initial data were studied in the works [7,11,12,13,14]. The proofs of the theoretical results were confirmed numerically in the works [17,18]. The exact solutions of the systems are presented in [2,3,4,5,9,10,15,16].…”

On the Rate of Stabilization of Solutions to the Cauchy Problem for the Godunov–Sultangazin System with Periodic Initial Data

“…In later works, a review of kinetic equations was implemented by V. V. Vedenyapin [12], O. A. Vasil'eva [9], E. V. Radkevich [6,7], O. V. Il'in [4], and many other authors.…”

“…≤ q 2 , q 2 ∈ (0, 1), then there exists a unique solution Z (m) ∈ L 2,γ (R + ; H (m) σ ) of the nonlinear equation (9) satisfying the inequality…”