Denise Aregba-Driollet

Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whithams stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems

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.

A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis

We introduce a degenerate nonlinear parabolic system that describes the chemical aggression of calcium carbonate stones under the attack of sulphur dioxide. For this system, we present some finite element and finite difference schemes to approximate its solutions. Numerical stability is given under suitable CFL conditions. Finally, by means of a formal scaling, the qualitative behavior of the solutions for large times is investigated, and a numerical verification of this asymptotics is given. Our results are in qualitative agreement with the experimental behavior observed in the chemical literature.

Asymptotic High-Order Schemes for

We investigate finite difference schemes which approximate

2\times2

2\times 2

Dissipative Hyperbolic Systems

one-dimensional linear dissipative hyperbolic systems. We show that it is possible to introduce some suitable modifications in standard upwinding schemes, which keep into account the long-time behavior of the solutions, to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states, respectively, around stationary solutions and in the diffusion (Chapman-Enskog) limit.

Kinetic approximation of a boundary value problem for conservation laws

Summary.We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.

Time asymptotic high order schemes for dissipative BGK hyperbolic systems

We introduce a new class of finite differences schemes to approximate one dimensional dissipative semilinear hyperbolic systems with a BGK structure. Using precise analytical time-decay estimates of the local truncation error, it is possible to design schemes, based on the standard upwind approximation, which are increasingly accurate for large times when approximating small perturbations of constant asymptotic states. Numerical tests show their better performances with respect to those of other schemes.

Diffusive Discrete BGK Schemes for Nonlinear Hyperbolic-parabolic Systems

We design numerical schemes for nonlinear degenerate parabolic systems. These schemes are based on discrete BGK models where both characteristic velocities and the source-term depend singularly on the relaxation parameter. Although we restrict this paper to cartesian grids, the method may be also adapted to unstructured meshes.