Catálogo de publicaciones - libros

We study the stability of travelling wall profiles for a one dimensional model of ferromagnetic nanowire submitted to an exterior magnetic field. We prove that these profiles are asymptotically stable modulo a translation-rotation for small applied magnetic fields.

Maxwell equations are easily resolved when the computational domain is convex or with a smooth boundary, but if on the contrary it includes geometrical singularities, the electromagnetic field is locally unbounded and globally hard to compute. The challenge is to find out numerical methods which can capture the EM field accurately. Numerically speaking, it is advised, while solving the coupled Maxwell-Vlasov system, to compute a continuous approximation of the field. However, if the domain contains geometrical singularities, continuous finite elements span a strict subset of all possible fields, which is made of the -regular fields. In order to recover the total field, one can use additional ansatz functions or introduce a weight. The first method, known as the singular complement method [4, 3, 14, 2, 9, 15, 16] works well in 2 and 2½ geometries and the second method, known as the weighted regularization method [13] works in 2 and 3. In this contribution, we examine some recent developments of the latter method to solve instationary Maxwell equations and we provide numerical results.

Wave equations are especially challenging for numerical integratorss since the solution is often not smooth and there is no smoothing in time. The largest usable step size of standard integrators, as for example the often used Störmer- Verlet-Leap-Frog-scheme, depends on the space discretisation. The better the approximation in space, the smaller the required step size of the integrator. The presented exponential integrator allows for error bounds independent of the space discretisation but only dependent on constants arising from the original problem. This favourable property is demonstrated with the Sine-Gordon equation.

In this paper, we consider exponential integrators that are based on linear multistep methods and study their positivity properties for abstract evolution equations. We prove that the order of a positive exponential multistep method is two at most and further show that there exist second-order methods preserving positivity.

The immersed boundary method is both a mathematical formulation and a numerical method for the study of fluid structure interactions. Many numerical schemes have been introduced to reduce the difficulties related to the non-linear coupling between the structure and the fluid evolution; however numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates for the variational formulation of the immersed boundary method.

In optical rewritable recording media, such as the Blu-ray Disc, amorphous marks are formed on a crystalline background of a phase-change layer, by means of short, high power laser pulses. It is of great importance to understand the mark formation process, in order to improve this data storage concept. The recording layer is part of a grooved multi-layered geometry, consisting of a variety of materials of which the material properties are assumed to be constant per layer, but may differ by various orders of magnitude in different layers. The melting stage of the mark formation process requires the inclusion of latent heat. In this study a comparison is made of numerical techniques for resolving the associated Stefan problem. The considered methods have been adapted to be applicable to multi-layers.

In this paper we consider a classical finite difference approximation of the heat equation. We study the long time behaviour of the solutions of the considered scheme and various questions related to the fundamental solutions. Finally we obtain the first term in the asymptotic expansion of the solutions.

A numerical method for a 2-dimensional surface fire model taking into account moisture content and radiation is developed. We consider the combustion of a porous solid, where a simplified energy conservation equation is applied. The effects of the moisture content and the endothermic pyrolysis of the vegetation are introduced in the model by means of a multivalued function representing the enthalpy. Its resolution is based on the Yosida approximation of a perturbation of this operator. The radiation term allows us to cope with wind and slope effects. In order to avoid heavy time consuming computations, this term is approximated using the characteristic method, combined with a discrete ordinate method. Finally, the approximate solution of the energy equation in the porous solid is obtained using a finite element method together with a semi-implicit Euler algorithm in time.

For the radiosity equation, we investigate iterative solutions and acceleration by the Fast Multipole Method (FMM). In this paper, a new FMM for general kernels is proposed to solve this equation, inspired by the method proposed by Gimbutas and Rokhlin [SIAM J. Sci. Comput. 24, 796–817 (2002)].

We consider using a modified asymptotic procedure for the numerical modelling of various kinetic equations.