Transport-Equilibrium Schemes for Computing Nonclassical Shocks. Scalar Conservation Laws
Abstract: This paper presents a very efficient numerical strategy for computing the weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of either concave-convex or convex-concave flux functions. In such a situation, nonclassical shocks violating the classical Oleinik entropy criterion must be taken into account since they naturally arise as limits of certain diffusive-dispersive regularizations to hyperbolic conservation laws. Such discontinuities play an important part in the resolution of the Riemann problem and their dynamics turns out to be driven by a prescribed kinetic function which acts as a selection principle. It aims at imposing the entropy dissipation rate across nonclassical discontinuities, or equivalently their speed of propagation. From a numerical point of view, the serious difficulty consists in enforcing the kinetic criterion, that is in controling the numerical entropy dissipation of the nonclassical shocks for any given discretization. This is known to be a very challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic criterion at the discrete level. The resulting scheme provides in addition sharp profiles. Numerical evidences illustrate the validity of our approach.