We show how to derive time implicit formulations of relaxation schemes for the Euler equations for real materials in several space dimensions. In the fully time explicit setting, the relaxation approach has been proved to provide efficient and robust methods. It thus becomes interesting to answer the open question of the time implicit extension of the procedure. A first natural extension of the classical time explicit strategy is shown to fail in producing discrete solutions which converge in time to a steady state. We prove that this first approach does not permit a proper balance between the stiff relaxation terms and the flux gradients. We then show how to achieve a well-balanced time implicit method which yields approximate solutions at a perfect steady state.
Introduction.This work is devoted to the study of time implicit formulations of relaxation schemes for the Euler equations with general pressure laws.Over the past decade, relaxation schemes have received considerable attention. Such schemes are primarily intended to approximate the solutions of highly nonlinear hyperbolic systems. The design principle of relaxation schemes consists in approximating the solutions (say, the Riemann solutions) of a given highly nonlinear system by the solutions of a larger but weakly nonlinear system with singular perturbations. These perturbations take the form of stiff relaxation source terms which restore the algebraic nonlinearities of the original PDE model in the regime of an infinite relaxation parameter. Here the key issue is that these source terms must of course facilitate the derivation of the approximation procedure together with its nonlinear stability analysis as well.At the theoretical level, relevant relaxation methods have been proved to obey in their time explicit formulation several important stability properties ranging from L 1 stability (the phase space is preserved) to nonlinear stability like entropy inequalities; see [3], [6], [5], [15], [9], and the references therein. Moreover, some of these methods [3], [6] also enjoy accuracy properties like the exact capture of stationary contact discontinuities. From a numerical point of view, the simplicity in the time explicit formulation of these methods guarantees a very low computational effort. In addition, the property that their upwind mechanism stays virtually free from the exact pressure law makes them very useful in practice. At last, the relaxation strategy allows for a fruitful reinterpretation of some of the prominent approximate Riemann solvers as underlined by Bouchut [3] and also Leveque and Pelanti [17]. Such a reintrepretation
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