Dans cette thèse nous nous intéresserons aux équations aux dérivées partielles stochastiques (EDPS) d'un point de vue aussi bien théorique que numérique. Ces équations peuvent être vues comme une généralisation du concept d'équations aux dérivées partielles (EDP) déterministes, équations donnant des modèles dans de nombreux domaines tel que la physique, la biologie ou encore l'économie. L'aspect stochastique apparaît avec la volonté de prendre en compte des données que l'on ne connaît pas de façon déterministe et dont nous avons uniquement des informations statistiques. Ces données peuvent être aussi bien un coefficient de l'équation qu'un terme de force, on qualifie alors ces données de "bruits". De par leurs complexités, il est courant de ne pas avoir de solution formelle pour certaines EDPS, la résolution numérique est alors l'unique moyen d'obtenir certaines statistiques de la solution inconnues formellement. La discrétisation de cette source d'information représentée par les termes de bruit pose le problème de leur troncature. L'information contenue dans ces termes de bruits est infini, ainsi tout comme il est impossible de représenter numériquement, sauf cas particulier, de façon exacte une fonction sur l'intervalle $[0,1]$, il est impossible de stocker de façon exacte ces termes de bruits, se pose alors la question du traitement numérique de ces termes de bruits. Une des méthodes consiste à simuler le bruit afin d'obtenir une famille de trajectoires du bruit et résoudre pour chacune de ces trajectoires l'équation associée afin de pouvoir faire des statistiques sur l'ensemble des solutions obtenues, cette méthode correspond à la méthode de Monte-Carlo. Elle offre l'avantage d'être relativement simple à mettre en œuvre mais se pose alors des problèmes de lenteur de convergence dûs au coût unitaire des intégrations numériques de chaque trajectoire qui dépendent en général de la méthode déterministe utilisée, de la dimension du problème et de la variance des moments que l'on souhaite estimer. Une deuxième philosophie est la décomposition du bruit sur une base polynomiale adaptée à une mesure de référence (ici la mesure de Wiener). C'est la méthode principalement étudiée dans cette thèse. Nous décrirons comment à l'aide d'une décomposition dite en chaos il est possible d'obtenir des statistiques de solutions d'EDPS, mais également comment on peut se servir d'une telle décomposition afin de réduire la variance dans une méthode de Monte Carlo / In this thesis, we will be interested by the numerical approximation of SPDEs. Such equations can be seen as generalization of deterministic PDEs whose coefficients have been perturbed in order to take into account incertainties. Usually those incertainties are only known through their statistical properties. This kind of data could be included into the coefficients of the PDE or can be modelized through an infinite dimensional diffusion term in the second member. The main purpose of the numerical investigations concerning SPDEs is the estimation of the joint probability distribution of its solution, and practically the estimation of some moments or some event's probabilities. The discretization of the noise's information in the small scales implies a large number of additionnal parameters and yields, in general, problems. The first and most popular method used usually is the Monte Carlo method. It relies upon the simulation of a large number of trajectories of the noise followed by the numerical integration of the associated SPDE's solution. Its main advantage is its simplicity and its capacity to be parallelized. Nevertheless, its main drawback is the rather slow convergence due to the unit cost of numerical integration of each trajectory which depend on the deterministic method used, the problem's dimension. Also the convergence can be slowed down because of the large variance of the statistical moments we want to estimate. A second approach consists in the chaos expansion of the coefficients based on a reference measure (Wiener's mesure e.g.). It will be the main purpose of this thesis. We will describe how such an expansion can be made possible in the SPDEs' framework, through the examples of the KdV and Burgers stochastic equations, in order to obtain statistical moments of the solutions but also in order to reduce wariance within a Monte Carlo method

No description available.

by Li Kam Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaf 65). / Chapter CHAPTER 1 --- Soliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §1.1 --- Introduction --- p.4 / Chapter §1.2 --- Inverse Scattering --- p.6 / Chapter §1.3 --- N-Soliton Solution --- p.11 / Chapter §1.4 --- One-Soliton Solutions --- p.15 / Chapter §1.5 --- Two-Soliton Solutions --- p.18 / Chapter §1.6 --- Oscillating and Asymptotically Standing Solitons --- p.23 / Chapter CHAPTER 2 --- Asymptotic Behaviour of Nonsoliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §2.1 --- Introduction --- p.31 / Chapter §2.2 --- Main Results --- p.36 / Chapter §2.3 --- Lemmas --- p.39 / Chapter §2.4 --- Proof of the Main Results --- p.59 / References --- p.65

Zhou Yong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 51-53). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.2 / Chapter 2 --- Preliminaries --- p.10 / Chapter 3 --- Periodic Case --- p.22 / Chapter 4 --- Non-periodic Case --- p.35 / Bibliography --- p.51

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

published_or_final_version / Physics / Master / Master of Philosophy

The first main focus in the present project was to analyse the boundary treatment of the linearised Korteweg-de Vries equation. The second main focus was to derive a stable numerical solution using a high-order finite difference method. Since the model involved a third derivative in space, the numerical treatment of the boundaries was highly nontrivial. To aid the boundary treatment high-order accurate first and third derivative finite difference operators were employed. The boundaries are based on the summation-by-parts (SBP) framework, thereby guaranteeing linear stability. The boundary conditions were imposed using a penalty technique. A convergence study was performed where the derived numerical solution was compared with an analytical one. Fourth order accurate Runge-Kutta was used to time-integrate the numerical approximation. Measuring the rate of convergence, q, yielded q = 4 for 4th order accurate SBP-operators and q = 5.5 for 6th order accurate SBP-operators. Thus the convergence study proved the accuracy and stability of the numerical solution derived with the SBP-methodology.

Universiẗat-Gesamthochsch., Diss.--Paderborn, 1997.

Thesis (M.S.)--University of North Carolina at Wilmington, 2006. / Includes bibliographical references (leaves: [73]-74)

The physicists and mathematicians have put a lot of efforts in the numerical analysis of various types of partial differential equations on unbounded domain. The time- dependent partial differential equations(PDEs) also have a wide range of applications in physics, geography and many other interdisciplines. This thesis is concerned with the numerical solutions of such kind of partial differential equations on unbounded spatial domain, especially the Korteweg-de Vries(KdV) equations. Since it is unable to solve the problem directly due to its unboundedness, the common way to surpass such difficulty is to introduce proper conditions on the truncated artificial boundaries and to approximate the problem on a bounded domain, which is also known as the Absorbing Boundary Conditions(ABCs). One of the main contributions of this thesis is to design accurate local absorbing boundary conditions for linearized KdV equations and to extend the method to non- linear KdV equations on unbounded domain. Pad´e approximation is the main tool to approximate the cubic root in the construction of local absorbing boundary conditions(LABCs) for a linearized KdV equation on unbounded domain. Besides, we also introduce the continued fraction method in the approximation of cubic root. To avoid the high-order derivatives in the absorbing boundary conditions, a sequence of auxiliary variables are applied accordingly. Then the original problem on unbounded domain is reduced to an approximated initial boundary value(IBV) problem defined on a finite domain. Based on previous work, we are able to extend the method to the design of efficient local absorbing boundary conditions for nonlinear KdV equations on unbounded domain. The unifying approach method is applied to this nonlinear case. The idea of the unifying approach method is to separate inward- and outward-going waves and to build suitable approximated linear operator with a “one-way operator”. Then we unite the approximated linear operator with the nonlinear subproblem and propose boundary conditions for the nonlinear subproblem along the artificial boundaries. The numerical simulations are given to demonstrate the effectiveness and accuracy of our local absorbing boundary conditions. Keywords: Korteweg-de Vries equation; Local absorbing boundary conditions; Pad´e approximation; Continued fraction method; Unifying approach.