Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods

This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear

Systems of Differential-Algebraic Equations Encountered in the Numerical Modeling of High-Temperature Superconductors

RÉSUMÉ L’objectif principal de ce mémoire est d’étudier les systèmes d’Équations Différentielles et Algébriques (EDA) qui apparaissent lors de la modélisation numérique d'équipements électriques supraconducteurs à Haute Température Critique (HTC). Ces systèmes d’équations ainsi que le comportement non linéaire des matériaux supraconducteurs sont possiblement responsables des difficultés rencontrées lors de simulations numériques de ces appareillages. Dans la littérature, beaucoup d’attention a été portée aux problèmes liés à la nonlinéarité des matériaux, mais, au meilleur de notre connaissance, aucune étude des systèmes d'équations différentielles et algébriques n'a été répertoriée. Ainsi, il est essentiel d’approfondir nos connaissances à leur sujet dans le cadre de la simulation numérique d’équipements supraconducteurs à HTC. Dans ce document, nous présentons une revue de la théorie des supraconducteurs de type I et de type II. Cette revue nous permet de bien comprendre le potentiel des supraconducteurs à HTC en électrotechnique. Ces derniers se démarquent notamment par leur capacité à opérer en fort champ et par leur température critique élevée. Nous discutons que la simulation numérique d’équipements supraconducteurs permet de les optimiser à faible coût en améliorant certaines caractéristiques d'opération tel que les pertes en courant alternatif. Ensuite, nous présentons les principaux modèles physiques utilisés pour modéliser les équipements supraconducteurs. Plus précisément, nous décrivons un modèle 1-D utilisant une formulation en flux magnétique. Ce modèle est relativement simple mais son équation aux dérivées partielles possède une solution analytique connue. Ce modèle est donc utile pour s’introduire à la discipline et vérifier une méthode numérique implémentée dans un code. Puis, nous présentons des modèles 2-D et 3-D qui utilisent la formulation en champs magnétique. Ces modèles sont une meilleure approximation de la réalité que le modèle 1-D. Ils peuvent notamment considérer des matériaux de différentes natures et géométries. Cependant, ils sont plus complexes. Finalement, nous présentons un modèle qui utilise la formulation en potentiel vecteur magnétique sous sa forme intégrale. Ce modèle peut tenir compte d’effets 3-D en utilisant la bonne définition pour l’intégrale du potentiel vecteur. Nous présentons deux méthodes numériques pour discrétiser les équations des modèles physiques dans l'espace, soit la Méthode des Éléments Finis (MEF) et la Méthode Semi-Analytique (MSA). Nous montrons que la MEF est utilisée pour discrétiser une forme faible des équations à l'aide d'une approximation discrète de la solution sur un maillage constitué d'éléments. Nous introduisons deux types d'éléments: les éléments finis nodaux et les éléments d'arrête (edge elements). Finalement, nous présentons brièvement la MSA qui est utilisée pour discrétiser dans l'espace les équations de la formulation en potentiel vecteur magnétique sous sa forme intégrale. Cette méthode consiste à trouver une expression analytique reliant des champs et des potentiels aux termes sources sur une certaine discrétisation puis à résoudre le système d'équations résultant numériquement. Il s’agit d’une méthode à collocation par point.----------ABSTRACT The main objective of this thesis is to study the systems of Differential-Algebraic Equations (DAE) encountered in the numerical modeling of electrical devices made of High-Temperature Superconductors (HTS). These systems of equations and the nonlinear behavior of HTS are possibly responsible for the difficulties faced when simulating HTS devices. In the literature, much attention is paid to the issues related to the nonlinearity of HTS but, to the best of our knowledge, there is no in-depth study of the problems related to the systems of DAE. Consequently, it is essential to improve our knowledge about those systems, in the context of HTS modeling. In this document, we review the theory of type I and type II superconductors. This review is useful to understand the potential of HTS materials for power engineering applications. Their potential is mainly due to their ability to operate in strong fields and their high critical temperatures. We discuss that numerical simulation can be used to optimize HTS devices at low cost, by improving some quantities of interest, e.g.\ AC losses. We introduce the main physical models used for the modeling of HTS devices. We describe a 1-D model based on a magnetic flux density formulation. This model is relatively simple but has a known analytical solution for a nonlinear HTS problem. It is convenient to use as an introduction to the methodology used in this thesis and to verify a code. Then, we introduce a 2-D and a 3-D model based on a magnetic field formulation. These models provide a better representation of the reality than the 1-D model. They can consider materials with different properties and complex geometries. However, they are more complicated than the 1-D model. Finally, we review a model based on a magnetic vector potential formulation in integral form (A-V). This model can take into account 3-D effects by using the proper definition for the magnetic vector potential integral. We summarize two numerical methods to discretize the equations of the physical models in space, i.e.\ the Finite Element Method (FEM) and the Semi-Analytical Method (SAM). The FEM is used to discretize a weak form of the equations of the models using a discrete approximation of the solution over a mesh made of geometrical elements. We introduce two types of elements: nodal elements and edge elements. Then, we review the SAM, a numerical method used to discretize the equations of the magnetic vector potential formulation in integral form. It is a collocation method. We introduce systems of DAE. These systems of equations are obtained after discretizing the equations of the physical models in space. We discuss that the mathematical structure of a system of DAE can be described by a notion called the index. The index is the number of derivation required for a system of DAE to become a system of Ordinary Differential Equations (ODEs). We note that systems of DAE of index 2 in Hessenberg form are recurrent in variational problems. Subsequently, we discuss three strategies to discretize systems of DAE in time, i.e.\ direct discretization, reduction of the index and reformulation into semi-explicit form. The direct discretization strategy consists in applying directly an implicit time integration scheme to a system of DAE without reducing its index. In most cases, this yields a system of nonlinear equations. The reduction of index consists in reducing the index of the system of DAE and then reassess its structure. Systems of DAE of index 0 can be reformulated into a semi-explicit form and then discretized using an explicit method. We introduce two time transient solvers that use the direct discretization strategy, i.e. Differential-Algebraic System SoLver (DASSL) and Implicit Differential-Algebraic Solver (IDAS)

Stability criteria for nonlinear fully implicit differential-algebraic systems

This thesis contributes to the qualitative theory of differential-algebraic equations(DAEs) by providing new stability criteria for solutions of a class of nonlinear, fully implicit DAEs with a properly stated derivative term and tractability index one and two. A generalization of the Andronov-Witt Theorem addressing orbital stability is proved. To this purpose, a state space representation of differential-algebraic systems based on the tractability index is developed which has advantageous properties, e.g. moderate smoothness requirements, commutativity with linearization and an autonomous structure in case of autonomous DAEs. It allows a suitable definition of characteristic multipliers referring to the inherent dynamics, but given in terms of the DAE. Furthermore, the fundamentals of Lyapunov's direct method with respect to diffe- rential-algebraic systems are worked out. Novel denitions of Lyapunov functions for differentiable solution components of a DAE are stated, where the monotoni- cally decreasing total time derivative of a Lyapunov function along DAE solutions is expressed in terms of the original system. The topology of the domain of the inherent dynamics turns out to be decisive for nonlocal existence of solutions given a Lyapunov function. As a result, practical stability criteria for bounded solutions of autonomous DAEs and for general solutions of DAEs with bounded partial derivatives of the constitutive function arise. Known contractivity denitions for DAEs can be interpreted in the context of this approach

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Inverse dynamics of underactuated flexible mechanical systems governed by quasi-linear hyperbolic partial differential equations

Diese Arbeit befasst sich mit der inversen Dynamik unteraktuierter, flexibler, mechanischer Systeme, welche durch quasi-lineare hyperbolische partielle Differentialgleichungen beschrieben werden können. Diese Gleichungnen, sind zeitlich veränderlichen Dirichlet-Randbedingungen unterworfen, welche durch unbekannte, räumlich disjunkte, also nicht kollokierte Neumann-Randbedingungen erzwungen werden. Die zugrundeliegenden Gleichungen werden zunächst abstrakt hergeleitet, bevor verschiedene mechanische Systeme vorgestellt werden können, die mit der eingangs postulierten Formulierung übereinstimmen. Hierzu werden geometrisch exakte Theorien hergeleitet, welche in der Lage sind große Bewegungen schlanker Strukturen wie Seile und Balken, aber auch ganz allgemein, dreidimensionaler Festkörper zu beschreiben. In der Regel werden Anfangs-Randwertprobleme, die in der nichtlinearen Strukturdynamik auftreten, durch Anwendung einer sequentiellen Diskretisierung in Raum und Zeit gelöst. Diese Verfahren basieren für gewöhnlich auf einer räumlichen Diskretisierung mit finiten Elementen, gefolgt von einer geeigneten zeitlichen Diskretisierung, welche meist auf finiten Differenzen beruht. Ein kurzer Überblick über derartige sequentielle Integrationsverfahren für das vorliegende Anfangs-Randwertproblem wird zunächst anhand der direkten Formulierung des Problems gegeben werden. D.h. es wird zunächst das reine Neumann-Randproblem betrachtet, bevor anschließend ganz allgemein, verschiedene Möglichkeiten zur Einbindung etwaiger Dirichlet-Randbedingungen diskutiert werden. Darauf aufbauend wird das Problem der inversen Dynamik im Kontext räumlich diskreter mechanischer Systeme, welche rheonom-holonomen Servo-Bindungen unterliegen, eingeführt. Eine ausführliche Untersuchung dieser Art von gebundenen Systemen soll die grundlegenden Unterschiede zwischen Servo-Bindungen und klassischen Kontakt-Bindungen herausarbeiten. Die daraus resultierenden Folgen für die Entwicklung geeigneter numerisch stabiler Integrationsverfahren können dabei ebenfalls angesprochen werden, bevor zahlreich ausgewählte Beispiele vorgestellt werden können. Aufgrund der sehr eingeschränkten Anwendbarkeit der sequentiellen Lösung der inversen Dynamik in Raum und Zeit, wird eine eingehende Analyse des vorliegenden Anfangs-Randwertproblems unternommen. Vor allem durch die Freilegung der hyperbolischen Struktur der zugrundeliegenden partiellen Differentialgleichungen werden sich weitere Einblicke in das vorliegende Problem erhofft. Die Erforschung der daraus resultierenden Mechanismen der Wellenausbreitung in kontinuierlichen Strukturen öffnet die Tür zur Entwicklung numerisch stabiler Integrationsverfahren für die inverse Dynamik. So kann unter anderem eine Methode vorgestellt werden, die auf der Integration der partiellen Differentialgleichungen entlang charakteristischer Mannigfaltigkeiten beruht. Dies regt zu der Entwicklung neuartiger Galerkinverfahren an, die ebenfalls in dieser Arbeit vorgestellt werden können. Diese neu entwickelten Methoden können anschlie\ss end auf die Steuerung verschiedener mechanischer Systeme angewendet werden. Darüber hinaus können die neuartigen Integrationsverfahren auch auf flexible Mehrkörpersysteme übertragen werden. Angeführt seien hier beispielsweise die kooperative Steuerung eines an mehreren flexiblen Seilen aufgehängten starren Körpers oder die Steuerung des Endeffektors eines flexiblen mehrgliedrigen Schwenkarms. Ausgewählte numerische Beispiele verdeutlichen die Relevanz der hier vorgeschlagenen, in Raum und Zeit simultanen Integration des vorliegenden Anfangs-Randwertproblems

On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context of the 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystr\"{o}m method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented

The Biglobal Instability of the Bidirectional Vortex

State of the art research in hydrodynamic stability analysis has moved from classic one-dimensional methods such as the local nonparallel approach and the parabolized stability equations to two-dimensional, biglobal, methods. The paradigm shift toward two dimensional techniques with the ability to accommodate fully three-dimensional base flows is a necessary step toward modeling complex, multidimensional flowfields in modern propulsive applications. Here, we employ a two-dimensional spatial waveform with sinusoidal temporal dependence to reduce the three-dimensional linearized Navier-Stokes equations to their biglobal form. Addressing hydrodynamic stability in this way circumvents the restrictive parallel-flow assumption and admits boundary conditions in the streamwise direction. Furthermore, the following work employs a full momentum formulation, rather than the reduced streamfunction form, accounting for a nonzero tangential mean flow velocity. This approach adds significant complexity in both formulation and implementation but renders a more general methodology applicable to a broader spectrum of mean flows. Specifically, we consider the stability of three models for bidirectional vortex flow. While a complete parametric study ensues, the stabilizing effect of the swirl velocity is evident as the injection parameter, kappa, is closely examined

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed