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

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Abstract
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.