We obtain sharp decay estimates in time in the context of Sobolev spaces for smooth solutions to the one-dimensional Jin Xin model under the diffusion scaling, which are uniform with respect to the singular parameter of the scaling. This provides the convergence to the limit nonlinear parabolic equation both for large time and for the vanishing singular parameter. The analysis is performed by means of two main ingredients.

We provide a framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin-Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers have been studied, and several integer sequences related to them have been introduced. In this article other types of Sheffer polynomials are considered, by introducing the adjoint Poisson-Charlier and the adjoint Related polynomials.

The use of Euler polynomials and Euler numbers allows us to construct a quadrature rule similar to the well-known Euler--MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval. An expression of the remainder term and numerical examples are also given. © 2001, Heldermann Verlag. All rights reserved.

A quadrature rule using Appell polynomials and generalizing both the Euler-MacLaurin quadrature formula and a similar quadrature rule, obtained in Bretti et al [15], which makes use of Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the extrema of the considered interval, is derived. An expression of the remainder term and a numerical example are also enclosed.

The paper deals with de la Vallee Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given.

Front propagation in two-dimensional steady and unsteady cellular flows is investigated in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. For the steady flow, a simplified model allows for an analytical prediction of the front speed v(f) dependence on the stirring intensity U, which is in good agreement with numerical estimates. In particular, at large U, the behavior v(f)similar toU/log(U) is predicted. By adding small scales to the velocity field we found that their main effect is to renormalize the flow intensity.

No abstract available

The study of the underlying physics of soft flowing materials depends heavily on numerical simulations, due to the complex structure of the governing equations reflecting the competition of concurrent mechanisms acting at widely disparate scales in space and time. A full-scale computational modelling remains a formidable challenge since it amounts to simultaneously handling six or more spatial decades in space and twice as many in time.