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DAMPING, STABILIZATION, AND NUMERICAL FILTERING FOR THE MODELING AND THE SIMULATION OF TIME DEPENDENT PDES

Abstract : 1. Introduction. One of the particularly hard issues in hydrodynamics is the modeling of damping phenomena: according to the physical situations, viscous (entire or fractional power of -Delta), local or non local additional terms (half-time or half-space derivative) have been proposed to represent the damping and the fitting with real physical data still remains a challenge, we refer the reader, e.g., to [56, 57]. The mathematical analysis of the long time behavior of the solutions of the resulting models is also essential to the understanding of the underlying physics [23, 41, 42, 43, 44, 47]. Of course the derivation of appropriate and robust numerical schemes is crucial to capture the dynamics and also to point out mathematical properties that are difficult to establish, [12, 21, 29, 39]. It is to be noticed that the presence of a damping term can be seen as a stabilization technique used in control theory, see [49, 58]. Let us look now to an apparently different topic: the conception of numerical solvers for nonlinear parabolic equations. It is a classical technique to enhance the We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, or for building adapted damping operators in the case of dispersive equation. We consider numerical filtering provided by mutigrid-like techniques as well as the filtering resulting from operator with monotone symbols. Our approach applies to several discretization techniques and we focus on finite elements and finite differences. Numerical illustrations are given on Cahn-Hilliard, Korteweig-de Vries and KuramotoSivashinsky equations.
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Soumis le : lundi 28 mars 2022 - 15:44:10
Dernière modification le : lundi 12 septembre 2022 - 11:32:27

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Jean-Paul Chehab. DAMPING, STABILIZATION, AND NUMERICAL FILTERING FOR THE MODELING AND THE SIMULATION OF TIME DEPENDENT PDES. Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 2021, 14 (8), pp.2693-2728. ⟨10.3934/dcdss.2021002⟩. ⟨hal-03621845⟩

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