Catálogo de publicaciones - libros

In this paper we discuss numerical method for a pore scale model for precipitation and dissolution in porous media.We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion. A semi-implicit time stepping is combined with a regularization approach to construct a stable and convergent numerical scheme. For dealing with the emerging time discrete nonlinear problems we propose here a simple fixed point iterative procedure.

We design finite volume schemes for two dimensional parabolic degenerate systems by using a kinetic, formally BGK, approach. The hyperbolic and parabolic parts are not splitted and the schemes are Riemann solver free. Moreover the spatial discretization can be written analytically, so that the implementation is easy. Some numerical tests are presented.

The purpose of this note is to review recent results by the authors on the well-posedness of entropy and renormalized entropy solutions for anisotropic doubly nonlinear degenerate parabolic equations.

A multiresolution method for a one-dimensional strongly degenerate parabolic equation modeling sedimentation-consolidation processes is introduced. The method is based on the switch between central interpolation or exact evaluation of the numerical flux combined with a thresholded wavelet transform applied to point values of the solution to control the switch. A numerical example is presented.

The aim of this paper is to collect some results concerning relaxation limits of hyperbolic systems of balance laws toward parabolic equilibrium systems. More precisely, we will discuss BGK approximations for strongly parabolic systems in the case of weak solutions, by means of compensated compactness techniques. Moreover, we will study the case of a semilinear relaxation approximation to a 2×2 hyperbolic-parabolic equilibrium system, with applications to viscoelasticity, in the case of classical solutions in one and several space variables. The latter case will be used as a case study to apply the modulated energy estimates.

It has been shown that the equation of diffusion, linear and nonlinear, can be obtained in a suitable scaling limit by a two-velocity model of the Boltzmann equation [7] . Several numerical approximations were introduced in order to discretize the corresponding multiscale hyperbolic systems [8, 1, 4]. In the present work we consider relaxed approximations for multiscale kinetic systems with asymptotic state represented by nonlinear diffusion equations. The schemes are based on a relaxation approximation that permits to reduce the second order diffusion equations to first order semi-linear hyperbolic systems with stiff terms. The numerical passage from the relaxation system to the nonlinear diffusion equation is realized by using semi-implicit time discretization combined with ENO schemes and central differences in space. Finally, parallel algorithms are developed and their performance evaluated. Application to porous media equations in one and two space dimensions are presented.

In this communication, presented in the minisymposium on Degenerated Parabolic Equations, we are interested in the mathematical analysis of a stratigraphic model concerning geologic basin formation. Firstly, we present the physical model and the mathematical formulation, which lead to an original degenerated parabolic - hyperbolic conservation law. Then, the definition of a solution and some mathematical tools in order to resolve the problem are given. At last, we present numerical illustrations in the 1 — case and we give some open problems.

We present a two-level non-overlapping additive Schwarz method for Discontinuous Galerkin approximations of elliptic problems. In particular, a two level-method for both symmetric and non-symmetric schemes will be considered and some interesting features, which have no analog in the conforming case, will be discussed. Numerical experiments on non-matching grids will be presented.

We deal with the numerical solution of a scalar nonstationary nonlinear convection-diffusion equation. We present a scheme which uses a discontinuous Galerkin finite element method for a space semi-discretization and the resulting system of ordinary differential equations is discretized by backward difference formulae. The linear terms are treated implicitly whereas the nonlinear ones by a higher order explicit extrapolation which preserves the accuracy of the schemes and leads to a system of linear algebraic equations at each time step. Thenumerical examples presented verify expected orders of convergence.

The paper is concerned with some aspects of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of convection-diffusion problems and compressible flow. In particular, theoretical analysis of the spacetime discontinuous Galerkin discretization is briefly discussed. The robustness of the DGFEM is demonstrated by its application to the simulation of compressible low Mach number flows.