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Habilitation à diriger des recherches

Inégalités d'Ingham et schémas semi-lagrangiens pour l'équation de Vlasov

Michel Mehrenberger 1, 2
1 CALVI - Scientific computation and visualization
IRMA - Institut de Recherche Mathématique Avancée, LSIIT - Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection, Inria Nancy - Grand Est, IECL - Institut Élie Cartan de Lorraine
Abstract : In the first part, we gather several results in the control theory around Ingham inequalities which are generalizations of Parseval's equality and appear for showing the internal or boundary observability, controlability or stabilization of the wave equation or similar equations in certain particular cases. We are interested at first in the optimality of such inequalities, by generalizing a previous result in the vec- torial case. We then develop a Ingham type theorem adapted to treat the case of a cartesian geometry. Finally, we give some observability results in the case of nume- rical approximations. In a second part, we present the semi-Lagrangian method which is composed by essentially two ingredients : the computation of the caracteristics along which the distribution function is constant et the interpolation step. We analyse high order schemes in time based on directional splitting, which are a succession of linear transport steps. We then study the semi-Lagrangian methods in this particular case and we make the link between different formulations. We also obtain a convergence theorem for the Vlasov-Poisson system in this framework, which remains valid in the case of small displacements. We then develop this type of methods in a more ge- neral framework, by using one dimensionnal conservative splitting. We also consider a discontinuous Galerkin variant of such schemes. In a last part, we study the gyroaverage operator which appears in plasma physics by taking care of finite Larmor radius corrections. Finally, we discuss the problematic of zero discrete divergence which gives a compatibility between field computations and the numerical method of transport.
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Habilitation à diriger des recherches
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Submitted on : Wednesday, September 26, 2012 - 10:22:54 PM
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Michel Mehrenberger. Inégalités d'Ingham et schémas semi-lagrangiens pour l'équation de Vlasov. Equations aux dérivées partielles [math.AP]. Université de Strasbourg, 2012. ⟨tel-00735678⟩



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