ENTROPY FLUX-SPLITTINGS FOR
            HYPERBOLIC  CONSERVATION LAWS
              (PART1:GENERAL FRAMEWORK)

    Gui-Quiang Chen
    Philippe G.LeFloch


RAPPORT NUMERO 302


A general framework is proposed for the derivation and analysis of 
flux-splittings and the corresponding flux-splitting schemes for systems 
of conservation laws endowed with a strictly convex entropy.
The approach leads to several new properties 
of the existing flux-splittings and to a method for the construction of 
entropy flux-splittings for general situations. 
A large family of genuine entropy flux-splittings is derived
for several significant examples: the scalar conservation laws, 
the p-system, and the Euler system of isentropic gas dynamics.
In particular, for the isentropic Euler system, we obtain a family of 
splittings that satisfy the entropy inequality associated with the 
mechanical energy. 
For this system, it is proved that there exists a unique genuine entropy  
flux-splitting that satisfies all of the entropy inequalities, 
which is also the unique diagonalizable splitting. 
This splitting can be also derived by the 
so-called kinetic formulation.
Simple and useful difference schemes are derived from
the flux-splittings for hyperbolic systems.
Such entropy flux-splitting schemes are shown to satisfy 
a discrete cell entropy inequality.    
For the diagonalizable splitting schemes, the principle of bounded invariant 
regions applies and provides an a priori $L^\infty$
estimate.   The convergence of entropy flux-splitting 
schemes is proved for the $2\times 2$ systems of conservation laws 
and the isentropic Euler system.

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