Computing undercompressive waves with the random choice scheme. Nonclassical shock waves

For several nonlinear hyperbolic models of interest we investigate the stability and large-time behavior of undercompressive shock waves characterized by a kinetic relation. The latter are considered as interfaces between two materials with distinct constitutive relations. In this first part we study nonclassical entropy solutions to scalar conservation laws with concave-convex flux-function and a non-genuinely nonlinear, strictly hyperbolic model of two conservation laws arising in nonlinear elastodynamics. We use Glimm's random choice scheme but we replace the classical Riemann solver with the {\it nonclassical\/} one described recently in \cite{\refLFlochfour, \refLTone}. Our numerical experiments demonstrate the robustness and accuracy of the random choice scheme for computing nonclassical shock waves which are known to be very sensitive to dissipation and dispersion mechanisms. In this paper, we study carefully various issues related to nonclassical shocks and their stability under perturbations. This numerical study yields important hints for further theoretical investigation on, for instance, the {\it double N-wave pattern \/} put forward when studying the time-asymptotic behavior of periodic nonclassical solutions.