On interface transmission conditions for conservation laws with discontinuous flux of general shape

Abstract

Conservation laws of the form ∂tu + ∂xf(x;u) = 0 with space-discontinuous flux f(x;⋅) = fl(⋅)1x<0 + fr(⋅)1x>0 were deeply investigated in the past ten years, with a particular emphasis in the case where the fluxes are "bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of fl,r. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers are then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications.