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

Boris Andreianov (1) , Clément Cancès (2)

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

We consider hyperbolic scalar conservation laws with discontinuous flux function of type $\partial_t u + \partial_x f(x,u) = 0$, with $f(x,u) = f_L(u) \Char_{\R^-}(x) + f_R(u) \Char_{\R^+}(x)$. Here $f_{L,R}$ are compatible bell-shaped flux functions as appear in numerous applications. It was shown in previous works 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 $f(u)|_{x=0}\leq \bar F$, first introduced by R. Colombo and P. Goatin. 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 $\bar F$.

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

Boris Andreianov, Clément Cancès. The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions. 2011. ⟨hal-00631586v1⟩