The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions

We consider hyperbolic scalar conservation laws with discontinuous flux function of the type ∂tu+∂xf(x,u)=0withf(x,u)=fL(u)1R−(x)+fR(u)1R+(x). Here, fL,RfL,R are compatible bell-shaped flux functions as appear in numerous applications. It was shown by Adimurthi and Gowda [S. Mishra Adimurthi, G.D.V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ. 2 (4) (2005) 783–837] and Bürger et al. [R. Bürger, K.H. Karlsen, J.D. Towers, An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal. 47 (3) (2009) 1684–1712] that several notions of solution make sense, according to a choice of the so-called (A,B)(A,B)-connection. In this note, we remark that every choice of connection (A,B)(A,B) corresponds to a limitation of the flux under the form f(u)∣x=0≤F̄, first introduced by Colombo and Goatin [R.M. Colombo, P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations 234 (2) (2007) 654–675. http://dx.doi.org/10.1016/j.jde.2006.10.014]. Hence we derive a very simple and “cheap to compute” formula for the Godunov numerical flux across the interface {x=0}{x=0}, for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter F̄. A numerical illustration is provided.