Global properties of a difference approximation of the free boundary problem for symmetric motion of a viscous barotropic gas

Abstract

We consider the initial boundary value problem for the system of quasilinear equations of symmetric motion of a viscous barotropic gas (a compressible fluid) with free boundary and study a special nonlinear two-level difference scheme. We obtain estimates of a difference solution, which are uniform in t > 0, and prove its stabilization as t —» +00 (with a stabilization rate estimate). The conditions for uniqueness of the solution as well as the scheme implementation by Newton's method are analysed. An extensive literature is devoted to the analysis of the solution estimates global in t > 0 and the solution behaviour as t —» + oo for equations of one-dimensional motion of a viscous gas without the assumption of data smallness [6,14,18] and [7-13, 21-24,26,27]. From the viewpoint of computational mathematics of great interest is the construction and the study of adequate difference approximations of the above equations such that difference solutions inherit the global properties of continuous ones [1,19,20,25]. Of importance is the development of techniques to study the global properties of the difference solutions. In this paper we study these problems for a special two-level difference scheme, which generalizes the scheme in [3,4] and [15], for equations of symmetric motion of a viscous barotropic gas with allowance for the mass force, given a free boundary. In this case we essentially use the methods of [20,24,27]. Moreover, we study the conditions for uniqueness of the solution to the difference scheme as well as its simple implementation by Newton's method. 1. INITIAL BOUNDARY VALUE PROBLEM. NOTATION AND AUXILIARY INFORMATION The symmetric motion of a viscous barotropic gas (a compressible fluid) in Lagrangian mass coordinates x, Hs described by the system of quasilinear equations D* = D(ru), η = l/p Dtu = Da + 0[r], σ = ν(η)ρϋη ρ(η) Dtr = u. Here η(χ,ί) > 0, u(x,i), r(x,t) are the sought functions, and Dt = d/dt, D = d/dx, g[r](x,t) =^(r(x, 2. The physical meaning of the values is: 77, u, r are the specific volume, the speed, the Euler coordinate; ρ, σ, ρ are the density, the stress, the pressure \p ρ(η) is an equation 'Moscow Power Engineering Institute (Technical University), Moscow 111250, Russia The work is supported by the Russian Foundation for the Basic Research (97-01-00214) and the Program ^Russian Universities-Basic Research*. Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/19/15 2:25 PM