Entropy Flux - Splittings

A general framework is proposed for the derivation and analysis of ux-splittings and the corresponding ux-splitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing ux-splittings and to a method for the construction of entropy ux-splittings for general situations. A large family of genuine entropy ux-splittings is derived for several signiicant 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 ux-splitting that satisses 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 diierence schemes are derived from the ux-splittings for hyperbolic systems. Such entropy ux-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 1 estimate. The convergence of entropy ux-splitting schemes is proved for the 2 2 systems of conservation laws and the isentropic Euler system.