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

Abstract

We consider hyperbolic scalar conservation laws with discontinuous flux function of the type ∂ t u + ∂ x f ( x , u ) = 0 with f ( x , u ) = f L ( u ) 1 R − ( x ) + f R ( u ) 1 R + ( x ) . Here, f L , 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 ) -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 ≤ F ̄ , 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 } , for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter F ̄ . A numerical illustration is provided.