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A version of the Glimm method based on generalized Riemann problems. (English) Zbl 1125.35003

This is an interesting theoretical foundation paper for the solution of the generalized Riemann problem using Glimm’s method. A number of theorems are presented for convergence, existence and uniqueness of the problem.
Authors’ summary: We introduce a generalization of Glimm’s random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.

MSC:

35A35 Theoretical approximation in context of PDEs
35L60 First-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
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