We consider the compressible Euler system of conservation laws for the mass, momentum, and energy of a gas. We prove that Murat and Tartar's compensated compactness method applies to this system in two cases: either (1) the equations of state have a special form (specifically, the Lagrangian sound speed is a function of the pressure only) that leads to the existence of a large family of entropy pairs or (2) the conservation law of energy is replaced by the conservation law for the specific entropy. Indeed, the latter is the physically meaningful formulation when the system is derived by relaxation from an isentropic two-phase mixture. The paper primarily investigates the structure of the gas dynamics system and includes a complete description of the (i) mathematical entropies, (ii) decoupling properties, (iii) invariant domains after Chuey, Conley, and Smoller's theory, and (iv) Tartar's commutation relations. The existence of weak solutions in cases (1) and (2) above is deduced from the pioneering work of DiPerna.

The paper is devoted to the construction of a higher order Roe-type numerical scheme for the solution of hyperbolic systems with relaxation source terms. It is important for applications that the numerical scheme handles both stiff and non stiff source terms with the same accuracy and computational cost and that the relaxation variables are computed accurately in the stiff case. The method is based on the solution of a Riemann problem for a linear system with constant coefficients: a study of the behavior of the solutions of both the nonlinear and linearized problems as the relaxation time tends to zero enables to choose a convenient linearization such that the numerical scheme is consistent with both the hyperbolic system when the source terms are absent and the correct relaxation system when the relaxation time tends to zero. The method is applied to the study of the propagation of sound waves in a two-phase medium. The comparison between our numerical scheme, usual fractional step methods, and numerical simulation of the relaxation system shows the necessity of using the solutions of a fully coupled hyperbolic system with relaxation terms as the basis of a numerical scheme to obtain accurate solutions regardless of the stiffness.

We consider a model of two-phase fluid flows composed of solid particles in a gas. This model writes in the form of an hyperbolic system of conservation laws with relaxation terms that account for the interphase drag force. The relaxation time tends to zero with the radius of the particles. In the limit of zero relaxation time, we replace the original model with stiff source terms by a relaxation system in the form of a convection-diffusion system. We are concerned with the propagation of acoustic waves. We give an empiric criterion to determine the range of validity of the relaxation model and to determine the range of relevance of the operator splitting method applied to the system with stiff relaxation terms.