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

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 diagonatizable 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 fluxsplitting schemes is proved for the 2 x 2 systems of conservation laws and the isentropic Euler system.

An Error Estimate for Finite Volume Methods for Multidimensional Conservation Laws

“…( 1 ) does not involve the modulus of continuity in time of the approximate solutions, (2) is based on the regularization (2.10) of the Kruzkov entropies, and (3) involves a piecewise constant approximation of the standard Kuznetsov test function, which is necessary to apply the discrete entropy inequalities derived in [9].…”

Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint

“…For a system with a strong entropy, for which we can prove a priori the condition 8) it gives a weak formulation of the problem thanks to the result. (2.4), (2.5), (2.7), with the condition (2.8), is a weak formulation for (1.4)- (1.6).…”

“…The first step of the scheme is suppose to be a flux-splitting scheme, as proposed in [8], and is written as…”

“…The flux of the system is here F (U ) = (ρu, ρu 2 + κρ γ ). We consider the flux-splitting scheme proposed in [8], which has the property to be entropy compatible. This scheme writes…”

“…We refer to [12] for the convergence to scalar equations, to [8,9] for the convergence to the system of Euler and to [17] for the convergence in the case of regular systems.…”