STABILITY AND CONVERGENCE OF THE DIFFERENCE SCHEMES FOR EQUATIONS OF ISENTROPIC GAS DYNAMICS IN LAGRANGIAN COORDINATES

Piotr Matus, Dmitry Polyakov

Abstract: For the the initial-boundary value problem (IBVP) for the isentropic gas dynamics written in Lagrangian coordinates in Riemann invariants we show how the necessary conditions for existence of global smooth solution can be obtained using technics due to P. Lax. Under these conditions we formulate the existence result in the class of piecewise-smooth functions. A priori estimates with respect to the input data for the difference scheme approximating this problem are obtained. Stability estimates are proved using only limitations for the initial and boundary conditions corresponding the differential problem. Estimates of stability in the general case have been obtained only for the finite instant of time $t<t_0$. The monotonicity is proved in both cases. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment illustrating theoretical results are given.

Keywords: Isentropic gas dynamics; Riemann invariants; Lagrangian coordinates; initial-boundary value problem; well-posedness and blow-up; finite difference scheme; stability

Classification (MSC2000): 65N06; 65M12

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