Convergence of a Godunov scheme for degenerate conservation laws with BV spatial flux and a study of Panov-type fluxes

Abstract

We prove the convergence of the Godunov-type scheme for a scalar conservation law in one space dimension with possibly infinitely many spatial discontinuities, which may have accumulation points. In contrast to the study appearing in Ghoshal, Jana and Towers [Convergence of a Godunov scheme to an Audusse–Perthame adapted entropy solution for conservation laws with BV spatial flux, Numer. Math. 146(3) (2020) 629–659], we do not restrict the flux to be unimodal and allow for the case where the flux has degeneracies due to which the corresponding singular map may not be invertible and hence, the so-called singular mapping technique is not applicable. We present two proofs of convergence depending on the nature of the flux. For fluxes which are piecewise constant in the space variable, convergence is established by proving [Formula: see text] bounds on the finite volume approximations away from the set of spatial discontinuity. On the other hand, if the fluxes are of Panov-type, i.e. [Formula: see text], we prove the convergence by establishing a novel [Formula: see text] property, which in turn implies the existence of [Formula: see text] bounds on the solution. We present numerical examples that illustrate the theory.