Claire David

On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation

In this paper, the compound Burgers–Korteweg–de Vries equation is studied by the first integral method, which is based on ring theory in commutative algebra. Several new kink-profile waves and periodic waves are established. The applications of these results to other nonlinear wave equations such as the modified Burgers–KdV equation and the compound KdV equation are discussed. The stability and bifurcations of the kink-profile waves are also indicated.

Structural stability of finite dispersion-relation preserving schemes

Abstract The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurrance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domains. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes [1] .

Spurious caustics of dispersion-relation-preserving schemes

A linear dispersive mechanism leading to a burst in the L ∞ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated with interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.

Control of the Black-Scholes equation

The purpose of this work is to apply the results developped by J.Y. Chemin and Cl. David, to the Black-Scholes equation. This latter equation being directly linked to the heat equation, it enables us to propose a new approach allowing to control properties of the solution by means of a shape parameter.

Symmetry Preserving Discretization of the Compressible Euler Equations

Symmetries are transformations which act on the physical variables of a system. They can transform the time, the position, the velocity and the thermodynamical properties (density, pressure) of the physical system. But they do not modify the evolution of the physical system. This work deals with continuous symmetries which are described by the Lie group theory. In physics, symmetries are space-time transformations, such as the Galilean transformations, the Lorentz transformations, the projective transformations, the scaling transformations, the translations.