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Well-posedness of the Riemann problem; consistency of the Godunov’s scheme. (English) Zbl 0685.65085
Current progress in hyperbolic systems: Riemann problems and computations, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Brunswick/ME (USA) 1988, Contemp. Math. 100, 251-265 (1989).
[For the entire collection see Zbl 0683.00014.]
Well-posedness of the Riemann problem (RP) and the consistency of the Godunov’s scheme for conservation laws that involves both hyperbolic and elliptic types is investigated. In particular the authors treat the equation for the motion of an elastic string \(u_ t=v_ x\), \(v_ t=(T(| u|)u/| u|)_ x\). The RP is proved in this case to be ill-posed both in sup-norm and in \(L_ 1\)-norm. As a consequence the Godunov’s scheme loses consistency with the original problem. Relaxing the string problem by convexifying the total energy provides a system which is well posed in \(L_ 1\), and for which then the consistency of the Godunov’s scheme holds. A number of numerical computations illustrate in an illuminating way the main points of the discussion.
Reviewer: Ph.Brenner

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
74B20 Nonlinear elasticity
Zbl 0683.00014