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 the type $$\partial_t u + \partial_x f(x,u) = 0 \text{\;\;with\;\;} f(x,u) = f_L(u) \Char_{\R^-}(x) + f_R(u) \Char_{\R^+}(x).$$ Here $f_{L,R}$ are compatible bell-shaped flux functions as appear in numerous applications. It was shown by Adimurthi, S. Mishra, G. D. V. Gowda ({\it J. Hyperbolic Differ. Equ. 2 (4) (2005) 783-837)} and R. Bürger, K. H. Karlsen and J. D. Towers ({\it SIAM J. Numer. Anal. 47~(3) (2009) 1684--1712}) that several notions of solution make sense, according to a choice of the so-called $(A,B)$-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. M. Colombo and P. Goatin ({\it J. Differential Equations 234 (2) (2007) 654-675}). Hence we derive a very simple and cheap to compute explicit formula for the Godunov numerical flux across the interface $\{x=0\}$, for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter $\bar F$. A numerical illustration is provided.

Mots clés: Scalar conservation laws; discontinuous flux functions; flux limitation; Godunov scheme; $L^1$ dissipative germs

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

Boris Andreianov, Clément Cancès. The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions. Applied Mathematics Letters, 2012, 25, pp.1844--1848. ⟨hal-00631586v2⟩