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

Boris Andreianov (1, 2) , Clément Cancès (3, 4)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Institut fuer Mathematik
3 RAPSODI - Reliable numerical approximations of dissipative systems
4 LJLL - Laboratoire Jacques-Louis Lions

Conservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x<0}+f_r(\cdot)\mathbf{1}_{x>0}$ 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: discontinuous flux; hyperbolic conservation law; monotone finite volume scheme; interface coupling; boundary layer; convergent scheme; well-posedness; entropy solution; interface flux

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. Journal of Hyperbolic Differential Equations, 2015, 12 (2), pp.343-384. ⟨10.1142/S0219891615500101⟩. ⟨hal-00940756v2⟩