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Numerical study of non-linear dispersive partial differential equations

Abstract : Numerical analysis becomes a powerful resource in the study of partial differential equations (PDEs), allowing to illustrate existing theorems and find conjectures. By using sophisticated methods, questions which seem inaccessible before, like rapid oscillations or blow-up of solutions can be addressed in an approached way. Rapid oscillations in solutions are observed in dispersive PDEs without dissipation where solutions of the corresponding PDEs without dispersion present shocks. To solve numerically these oscillations, the use of efficient methods without using artificial numerical dissipation is necessary, in particular in the study of PDEs in some dimensions, done in this work. As studied PDEs in this context are typically stiff, efficient integration in time is the main problem. An analysis of exponential and symplectic integrators allowed to select and find the more efficient method for each PDE studied. The use of parallel computing permitted to address numerically questions of stability and blow-up in the Davey-Stewartson equation, in both stiff and non-stiff regimes.
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Kristelle Vitali Roidot. Numerical study of non-linear dispersive partial differential equations. General Mathematics [math.GM]. Université de Bourgogne, 2011. English. ⟨NNT : 2011DIJOS041⟩. ⟨tel-00692445⟩

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