We introduce a general framework for kinetic BGK models. We assume to be given a system of hyperbolic conservation laws with a family of Lax entropies, and we characterize the BGK models that lead to this system in the hydrodynamic limit, and that are compatible with the whole family of entropies. This is obtained by a new characterization of maxwellians as entropy minimizers, that can take into account the simultaneous minimization problems corresponding to the family of entropies. We deduce a general procedure to construct such BGK models, and we give examples and applications in gas dynamics.

We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diffusive conservation laws. We prove the well-posedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution recently obtained by Carrillo. Examples of such BGK models are provided.

Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.

We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235--276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.

Contents 1 Introduction 2 2 Motivations 5 2.1 The basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The smooth case 14 3.1 Local smooth theory for quasi-linear hyperbolic systems with relaxation . . . 14 3.2 Stability of global simple waves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Discontinuous equilibrium solutions and weak convergence methods 26 4.1 A conservative framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Compensated compactness results . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Kinetic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 The BV framework 39 5.1 Weakly coupled systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2 Relaxation limits for the Jin-Xin model and other discrete kinetic approximations . . . . . . . . . . . . . . . . . . . .

We consider the Cauchy problem for a general one dimensional nn hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to nd a set of general and realistic sucient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we rst propose a general framework to set this kind of problems, by using the so-called entropy variables. Therefore, we pass to prove some general statements about the global existence of smooth solutions, under dierent sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some re ned energy estimates and the use of a suitable version of the Kawashima condition. 1.

We establish forward and backward relations between entropy satisfying BGK models such as those introduced previously by the author and the first order flux vector splitting numerical methods for systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes. We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support. A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers. We deduce the construction of finite velocity relaxation systems for gas dyn...

We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the source-term depend singularly on the relaxation parameter. General stability conditions are derived, and convergence is proved to the entropy solutions for scalar equations.

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the L p theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.

In this paper, we prove a global error estimate for a relaxation scheme approximating scalar conservation laws. To this end, we decompose the error into a relaxation error and a discretization error. Including an initial error !(ffl) we obtain the rate of convergence of p ffl in L 1 for the relaxation step. The estimate here is independent of the type of nonlinearity. In the discretization step a convergence rate of p \Deltax in L 1 is obtained. These rates are independent of the choice of initial error !(ffl). Thereby, we obtain the order 1=2 for the total error.