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

TL;DR

This paper proves the convergence of fully-discrete numerical schemes for diffusive-dispersive conservation laws with discontinuous flux, showing that solutions may include nonclassical shocks depending on diffusion and dispersion parameters.

Contribution

It establishes the convergence of schemes for scalar conservation laws with discontinuous coefficients, highlighting the emergence of nonclassical shocks.

Findings

Convergence of schemes to solutions with discontinuous flux.

Limiting solutions may contain nonclassical undercompressive shocks.

Examples illustrate the dependence on diffusion and dispersion coefficients.

Abstract

We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show 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. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kruzkov-Oleinik entropy solutions, but contain nonclassical undercompressive shock waves.