Entropy flux-splittings for hyperbolic conservation laws part I: General framework

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, an a priori L∞ estimate is provided by applying the principle of bounded invariant regions. The convergence of entropy flux-splitting schemes is proved for the 2 × 2 systems of conservation laws and the isentropic Euler system. ©1995 John Wiley & Sons, Inc.

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