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Stabilisation de quelques équations d’évolution du second ordrepar des lois de rétroaction / Stabilization of second order evolution equations with dynamical feedbacksAbbas, Zainab 02 October 2014 (has links)
Dans cette thèse, nous étudions la stabilisation de certaines équations d’évolution par des lois de rétroaction. Dans le premier chapitre nous étudions l’équation des ondes dans R avec conditions aux limites dynamiques appliquées sur une partie du bord et une condition de Dirichlet sur la partie restante. Nous fournissons des conditions suffisantes qui garantissent une stabilité polynomiale en utilisant une méthode qui combine une inégalité d’observabilité pour le problème non amorti associé avec des résultats de régularité du problème non amorti. L’optimalité de la décroissance est montrée dans certains cas à l’aide des résultats spectraux précis de l’opérateur associé. Dans le deuxième chapitre nous considérons le système sur un domaine de Rd, d ≥ 2. On trouve des conditions suffisantes qui permettent la stabilité forte. Ensuite, nous discutons de la stabilité non uniforme ainsi que de la stabilité polynomiale. L’approche en domaine fréquentiel nous permet d’établir une décroissance polynomiale sur des domaines pour lesquels l’équation des ondes avec l’amortissement standard est exponentiellement ou polynomialement stable. Dans le troisième chapitre nous considérons un cadre général d’équations d’évolution avec une dissipation dynamique. Sous une hypothèse de régularité, nous montrons que les propriétés d’observabilité pour le problème non amorti impliquent des estimations de décroissance pour le problème amorti. / In this thesis, we study the stabilization of some evolution equations by feedback laws. In the first chapter we study the wave equation in R with dynamical boundary control applied on a part of the boundary and a Dirichlet boundary condition on the remaining part. We furnish sufficient conditions that guarantee a polynomial stability proved using a method that combines an observability inequality for the associated undamped problem with regularity results of the solution of the undamped problem. In addition, the optimality of the decay is shown in some cases with the help of precise spectral results of the operator associated with the damped problem. Then in the second chapter we consider the system on a domain of Rd, d ≥ 2. In this case, the domain of the associated operator is not compactly embedded into the energy space. Nevertheless, we find sufficient conditions that give the strong stability. Then, we discuss the non uniform stability as well as the polynomial stability by two methods. The frequency domain approach allows us to establish a polynomial decay on some domains for which the wave equation with the standard damping is exponentially or polynomially stable. Finally, in the third chapter we consider a general framework of second order evolution equations with dynamical feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We finally illustrate our general results by a variety of examples.
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Šíření tlakových pulsací v pružných plastových hadicích / Pressure pulsation propagation in elastic hosesČapoš, Eduard January 2020 (has links)
This thesis deals with propagation of pressure and flow pulsations, which are strongly affected by the tube flexibility. There are two mathematic models introduced, which are derived from basic physical relations. First model assumes velocity only in the axis direction. Second one assumes also non-zero radial velocity. Kelvin-Voigt model for viscoelasticity was used. Furthermore, experimental measurement was designed and evaluated. Measured data was used to calculate material properties. In addition, dynamic transfer was determined.
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Estimations de dispersion et de Strichartz dans un domaine cylindrique convexe / Dispersive and Strichartz estimates for the wave equation inside cylindrical convex domainsMeas, Len 29 June 2017 (has links)
Dans ce travail, nous allons établir des estimations de dispersion et des applications aux inégalités de Strichartz pour les solutions de l’équation des ondes dans un domaine cylindrique convexe Ω ⊂ R³ à bord C∞, ∂Ω ≠ ∅. Les estimations de dispersion sont classiquement utilisées pour prouver les estimations de Strichartz. Dans un domaine Ω général, des estimations de Strichartz ont été démontrées par Blair, Smith, Sogge [6,7]. Des estimations optimales ont été prouvées dans [29] lorsque Ω est strictement convexe. Le cas des domaines cylindriques que nous considérons ici généralise les resultats de [29] dans le cas où la courbure positive dépend de l'angle d'incidence et s'annule dans certaines directions. / In this work, we establish local in time dispersive estimates and its application to Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains Ω ⊂ R³ with smooth boundary ∂Ω ≠ ∅. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair, Smith, Sogge [6,7]. Optimal estimates in strictly convex domains have been obtained in [29]. Our case of cylindrical domains is an extension of the result of [29] in the case where the nonnegative curvature radius depends on the incident angle and vanishes in some directions.
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On Traveling Wave Solutions of Linear and Nonlinear Wave Models (Seeking Solitary Waves)Moussa, Mounira 02 June 2023 (has links)
No description available.
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Finita differensapproximationer av tvådimensionella vågekvationen med variabla koefficienter / Finite Difference Approximations of the Two-Dimensional Wave Equation with Variable CoefficientsBergkvist, Herman January 2023 (has links)
I [Mattson, Journal of Scientific Computing 51.3 (2012), s. 650–682] konstruerades partialsummeringsoperatorer för finita differensapproximationer av andraderivator med variabla koefficienter. Vi tillämpar framgångsrikt dessa operatorer på vågekvationen i två dimensioner med diskontinuerliga koefficienter, utan särskild behandling av diskontinuiteten. Närmare bestämt undersöks (i) operatorernas fel och konvergensordning relativt ”korrekt” hantering av diskontinuiteter genom blockuppdelning med kopplingstermer; (ii) ifall mycket komplicerade koefficienter orsakar instabilitet eller icke-fysikaliska fel. Vi visar att hoppet i våghastighet i simuleringen sker ett antal punkter ifrån hoppet i koefficienter, där antalet punkter beror på operatorernas ordning och storleken av hoppet i koefficienter. I (i) får dessa två faktorer plus blockets form och antalet punkter en stor påverkan på både storleken av felet, samt metodens konvergensordning som varierar från ca 1–2,5. Annars sker i både (i) och (ii) inget större icke-fysikaliskt fel eller instabilitet, vilket gör denna relativt enkla metod tillämpningsbar på komplexa verklighetsbaserade problem.
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Numerical simulation of shear instability in shallow shear flowsPinilla, Camilo Ernesto. January 2008 (has links)
No description available.
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Analysis and Implementation of High-Order Compact Finite Difference SchemesTyler, Jonathan G. 30 November 2007 (has links) (PDF)
The derivation of centered compact schemes at interior and boundary grid points is performed and an analysis of stability and computational efficiency is given. Compact schemes are high order implicit methods for numerical solutions of initial and/or boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique is employed. Experiments for non-stationary linear problems (convection, heat conduction) and also for nonlinear problems (Burgers' and KdV equations) were performed. The compact solvers were combined with Euler and fourth-order Runge-Kutta time differencing. In most cases, the order of convergence of the numerical solution to the exact solution was the same as the formal order of accuracy of the compact schemes employed.
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High-Accuracy and Stable Finite Difference Methods for Solving the Acoustic Wave EquationBoughanmi, Aimen January 2024 (has links)
This report presents a comprehensive investigation into the accuracy and stability of Finite Difference Methods (FDM) when applied to the acoustic wave equation. The analysis focuses on comparing the classical 2nd order FDM with highly-accurate computational stencil of order 2p = 2,4 and 6 with Summation-by-Parts (SBP) and Simultaneous Approximation Term (SAT) technique of the Finite Difference Method. The objective of the study is to investigate complex numerical techniques that contributes to highly-accurate and stable solutions to many hyperbolic PDEs. The report starts by introducing the governing problem and studies its well-posedness to ensure stable and unique solutions of the governing equations. It continues with basic introduction to the classic spatial discretization of the FDM and introduces the SBP-SAT implementation of the method. The governing equations are rewritten as a semi-discrete problem, such that it can be written as a system of ordinary differential equations (ODEs) only dependent on the temporal evolution. This system can be solved with classic Runge-Kutta methods to ensure robust and accurate time-stepping schemes. The results show that the implementation of the higher-order SBP-SAT Finite Difference Method provides highly accurate solutions of the acoustic wave equation compared to the classic FDM. The results also show that the method provides stable solutions with no visible oscillations (dispersion), which can be a challenge for higher order methods. Overall, this paper contributes with valuable insights into the analysis of accuracy and stability in finite difference methods for acoustic wave equation.
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Analýza disipativních rovnic v neomezených oblastech / Analysis of dissipative equations in unbounded domainsMichálek, Martin January 2013 (has links)
In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.
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Direct and Inverse scattering problems for elastic wavesXiaokai Yuan (6711479) 16 August 2019 (has links)
<p> In this thesis, both direct and inverse elastic scattering problems are considered. For a given incident wave, the direct problem is to determine the displacement of wave field from the known structure, which could be an obstacle or a surface in this thesis; The inverse problem is to determine the structure from the measurement of displacement on an artificial boundary.</p><p>In the second chapter, we consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.<br></p><p>In chapter 3, we extend the argument developed in chapter 2 to elastic surface grating problem, where the surface is assumed to be periodic and elastic rigid; Then, we treat the obstacle scattering in three dimensional space; The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Finally, in chapter 4, a rigorous mathematical model and an efficient computational method are proposed to solve the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangle slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near field data conversion and utilized the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem; Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction; Numerical examples are presented to demonstrate the effectiveness of the proposed methods for solving the questions mentioned above.<br></p>
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