ENTROPY FLUX-SPLITTINGS FOR HYPERBOLIC CONSERVATION-LAWS .1. GENERAL FRAMEWORK

A general framework is proposed for the derivation and analysis of flu x-splittings and the corresponding flux-splitting schemes for systems of conservation laws endowed with a strictly convex entropy. The appro ach leads to several new properties of the existing flux-splittings an d to a method for the construction of entropy flux-splittings for gene ral situations. A large family of genuine entropy flux-splittings is d erived for several significant examples: the scalar conservation laws, the p-system, and the Euler system of isentropic gas dynamics. In par ticular, for the isentropic Euler system, we obtain a family of splitt ings that satisfy the entropy inequality associated with the mechanica l energy. For this system, it is proved that there exists a unique gen uine entropy flux-splitting that satisfies all of the entropy inequali ties, which is also the unique diagonalizable splitting. This splittin g can be also derived by the so-called kinetic formulation. Simple and useful difference schemes are derived from the flux-splittings for hy perbolic systems. Such entropy flux-splitting schemes are shown to sat isfy a discrete cell entropy inequality. For the diagonalizable splitt ing schemes, an a priori L(infinity) estimate is provided by applying the principle of bounded invariant regions. The convergence of entropy flux-splitting schemes is proved for the 2 x 2 systems of conservatio n laws and the isentropic Euler system. (C) 1995 John Wiley & Sons, In c.