| keyword(s) : analyse numérique – systèmes de réaction-diffusion – méthodes de splitting – préconditionnement – méthodes spectrales – méthodes d'éléments finis – asymptotiques de polynômes – croissance de grains – systèmes auto-gravitants de fermions – méthode de la phase stationnaire – extrapolations de Richardson – théorie des semi-groupes. |
| Alternative location : http://maply.univ-lyon1.fr/~ribot/these.ps.gz http://maply.univ-lyon1.fr/~ribot/these.pdf |
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| We are interested in the study of numerical methods for reaction-diffusion systems. We first consider the Residual Smoothing Scheme and its extrapolations; this scheme uses a spatial preconditioner for the time discretization. We prove the stability of this method for the usual norm and its convergence in energy norm and we apply this scheme to the preconditioning of spectral methods by finite elements methods. For this application, we need to compute precise asymptotic formulas of Legendre polynomials and of their extrema. Then, we study a semi-discretization in time of a splitting scheme, called the Peaceman-Rachford approximation and we show that this scheme is convergent and of order two. Eventually, we apply these methods to the simulation of a parabolic non linear equation modelizing grain growth and to the computation of solutions of a non local parabolic equation coming from statistical mechanics and modelizing the fermionic self-gravitating systems. |
| english keyword(s) : numerical analysis – reaction-diffusion systems – splitting methods – preconditioning – spectral methods – finite elements methods – asymptotics of polynomials – grain growth – fermionic self-gravitating systems – stationary phase method – Richardson extrapolation |
