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

Boris Andreianov (1, 2) , Clément Cancès (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Institut fuer Mathematik
3 LJLL - Laboratoire Jacques-Louis Lions

Conservation laws of the form $\p_t u+ \p_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\1_{x<0}+f_r(\cdot)\1_{x>0}$ 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.

Mots clés: hyperbolic conservation law; discontinuous flux; interface coupling; boundary layer; monotone finite volume scheme; interface flux; entropy solution; well-posedness; convergent scheme

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Boris Andreianov, Clément Cancès. On interface transmission conditions for conservation laws with discontinuous flux of general shape. 2014. ⟨hal-00940756v1⟩