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) = 0 with f(x, u) = fL(u)1 R−(x) + fR(u)1 R+(x). Here fL,R are compatible bell-shaped flux functions as appear in numerous applications. In [1] and [2], it was shown that several notions of solution make sense, according to a choice of the so-called (A,B)-connection. In this note, we remark that every choice of connection (A,B) corresponds to a limitation of the flux under the form f(u)|x=0 ≤ F̄ , first introduced in [3]. Hence we derive a very simple and cheap to compute explicit formula for the the Godunov numerical flux across the interface {x = 0}, for each choice of the connection. This gives a simple-to-use numerical scheme governed only by the parameter F̄ . A numerical illustration is provided.