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

### Abstract

International audienceConservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x0}$ were deeply investigated in the last 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 $f_{l,r}$. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications

Topics: boundary layer, interface coupling, monotone finite volume scheme, discontinuous flux, hyperbolic conservation law, convergent scheme, well-posedness, entropy solution, interface flux, 35L65, 35L04, 35D30, 65N08, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Publisher: 'World Scientific Pub Co Pte Lt'
Year: 2015
DOI identifier: 10.1142/S0219891615500101
OAI identifier: oai:HAL:hal-00940756v2