Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient (Q2830699)

A finite difference method for the vanishing diffusive-dispersive approximations of scalar conservation laws with a discontinuous coefficient is presented. A fully-discrete finite difference scheme, in space and time, for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension is proposed. The convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient, is shown. The mathematical framework and the developed finite difference scheme are analyzed in detail. It is highlighted that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kružkov-Oleĭnik entropy solutions, but contain nonclassical undercompressive shock waves. Numerical experiments are used to illustrate the convergence and the performance of the developed scheme.