Shock capturing approximations to the compressible Euler equations with geometric structure and related equations (Q1392750)

A shock-capturing finite-difference scheme developed recently by \textit{G.-Q. Chen} and \textit{J. Glimm} [Commun. Math. Phys. 180, 153-193 (1996; Zbl 0857.76073)] is applied to construct approximate solutions to the Euler equations of isentropic gas dynamics. Using the compensated compactness arguments, the authors prove \(L^\infty\)-stability, \(H^1\)-compactness and convergence of the approximate difference solutions. Then they obtain global solutions for arbitrarily large \(L^\infty\) initial data. The possibilities of the new method are illustrated by applications to transonic nozzle flows, spherically symmetric flows, and to the piston problem.