A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In each of these classes, we present new A-stable schemes of orders six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as one that is stiffly accurate and/or L-stable. The latter require more stages but give better results for highly stiff problems and differential-algebraic equations (DAEs). The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and efficiency of the schemes are demonstrated on diverse problems.

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Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive. The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze. The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing. / Doctor of Philosophy / Almost all time-dependent physical phenomena can be effectively described via ordinary differential equations. This includes chemical reactions, the motion of a pendulum, the propagation of an electric signal through a circuit, and fluid dynamics. In general, it is not possible to find closed-form solutions to differential equations. Instead, time integration methods can be employed to numerically approximate the solution through an iterative procedure. Time integration methods are of great practical interest to scientific and engineering applications because computational modeling is often much cheaper and more flexible than constructing physical models for testing. Large-scale, complex systems frequently combine several coupled processes with vastly different characteristics. Consider a car where the tires spin at several hundred revolutions per minute, while the suspension has oscillatory dynamics that is orders of magnitude slower. The brake pads undergo periods of slow cooling, then sudden, rapid heating. When using a time integration scheme for such a simulation, the fastest dynamics require an expensive and small timestep that is applied globally across all aspects of the simulation. In turn, an unnecessarily large amount of work is done to resolve the slow dynamics. The goal of this dissertation is to explore new "multimethods" for solving differential equations where a single time integration method using a single, global timestep is inadequate. Multimethods combine together existing time integration schemes in a way that is better tailored to the properties of the problem while maintaining desirable accuracy and stability properties. This work seeks to overcome limitations on current multimethods, further the understanding of their stability, present new applications, and most importantly, develop methods with improved efficiency.

Earthquake ground motion can induce out-of-phase vibrations between adjacent structures due to differences in dynamic characteristics, which can result in impact or pounding of the structures if the at-rest separation is insufficient to accommodate the relative displacements. In bridges, seismic pounding between adjacent decks or between deck and abutment can result in localized deck damage, bearing failure, damage to shear keys and abutments, and even contribute to the collapse of bridge spans. This study investigates pounding in bridges from an analytical perspective. A simplified nonlinear model of a multiple-frame bridge is developed in MATLAB incorporating the effects of inelastic frame action, nonlinear hinge behavior and abutments. The equations of motion of the bridge response to longitudinal ground excitation are assembled and solved using the fourth-order Runge-Kutta method. Pounding is simulated using contact force-based models such as the linear spring, Kelvin and Hertz models, as well as the momentum-based stereomechanical method. In addition, a Hertz contact model with nonlinear damping (Hertzdamp model) is also introduced to model impact. The primary factors controlling the pounding response are identified as the frame period ratio, ground motion effective period ratio, restrainer stiffness ratio and frame ductility ratio. Pounding is most critical for highly out-of-phase frames. Impact models without energy dissipation overestimate the stiff system displacements by 15%-25% for highly out-of-phase, elastic systems experiencing moderate to strong ground excitation. The Hertzdamp model is found to be the most effective in representing impact. Traditional column hysteresis models such as the elasto-plastic and bilinear models underestimate the stiff system amplification and overestimate the flexible system amplification due to impact, when compared with stiffness and strength degrading models. Strength degradation and pounding are critical on the stiff system response to near field ground motions, for highly out-of-phase systems. Current design procedures are adequate in capturing the nonlinear hinge response when the bridge columns are elastic, but require revisions such as the introduction of time dependent reduction factors, and a frame design period to work for inelastic situations. Finally, a bilinear truss element with a gap is proposed for implementing energy dissipating impact models in commercial structural software.

L’algorithme covariant de van Holten, servant à construire des quantités conservées, est présenté avec une attention particulière portée sur les vecteurs de type Runge-Lenz. La dynamique classique des particules portant des charges isospins est passée en revue. Plusieurs applications physiques sont considérées. Des champs de type monopôles non-Abéliens,générés par des mouvements nucléaires dans les molécules diatomiques, introduites parMoody, Shapere et Wilczek, sont étudiées. Dans le cas des espaces courbes, le formalisme de van Holten permet de décrire la symétrie dynamique des monopôles Kaluza-Klein généralisés. La procédure est étendue à la supersymétrie et appliquée aux monopôles supersymétriques.Une autre application, concernant l’oscillateur non-commutatif en dimension trois, est également traitée. / Van Holten’s covariant algorithm for deriving conserved quantities is presented, with particular attention paid to Runge-Lenz-type vectors. The classical dynamics of isospin-carrying particles is reviewed. Physical applications including non-Abelian monopole-type systems in diatoms, introduced by Moody, Shapere and Wilczek, are considered. Applied to curved space, the formalism of van Holten allows us to describe the dynamical symmetries of generalized Kaluza-Klein monopoles. The framework is extended to supersymmetry and applied to the SUSY of the monopoles. Yet another application concerns the three-dimensional non-commutative oscillator.

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Neste trabalho, desenvolve-se um método de fronteira imersa para o estudo de escoamento compressível modelado pelas equações de Euler bidimensionais. O método de discretização de diferenças finitas é empregado, usando o método de Steger-Warming de ordem dois para discretizar as variáveis espaciais e o esquema de Runge-Kutta de ordem quatro para discretizar as variáveis temporais. O método da fronteira imersa foi empregado para o estudo de aeroelasticidade computacional em uma seção típica de aerofólio bidimensional com dois movimentos prescritos: torsional e vertical, com o objetivo de se verifcar a eficiência do método e sua aplicabilidade para problemas em aeroelasticidade computacional. Neste estudo desenvolveu-se também um programa de computador para simular escoamentos compressíveis de fluido invíscido utilizando a metodologia proposta. A verificação do código gerado foi feita utilizando o método das soluções manufaturadas e o problema de reflexão de choque oblíquo. A validação foi realizada comparando-se os resultados obtidos para o escoamento ao redor de uma seção circular e de uma seção de aerofólio NACA 0012 com os resultados experimentais, para cada caso. / In this work, an immersed boundary method is developed to study compressible flow modeled by the two-dimensional Euler equations. The finite difference method is employed, using the second order Steger-Warming method to discretizate the space variables and the fourth order Runge-Kutta method to discretizate the time variables. The immersed boundary method was employed to study computational aeroelasticity on a typical two-dimensional airfoil section with two prescribed motion: pitching and plunging, in order to verify the efficiency of the numerical method and its applicability in computational aeroelasticity problems. In this work, a computer program was developed to simulate compressible flows for inviscid fluids using the methodology proposed. The verification of the computational code was performed using the method of manufactured solutions and the oblique shock wave reflection problem. The validation was performed comparing the obtained results for flows around a circular section and a NACA 0012 airfoil section with the experimental results, for each case.

Schwerpunkt dieser Arbeit ist die Untersuchung einer Klasse von Diskretisierungsmethoden für nichtlineare optimale Steuerungsprobleme mit gewöhnlichen Differentialgleichungen und Steuerungsbeschränkung sowie die Durchführung von numerischen Experimente. Die theoretischen Untersuchungen basieren aus einem gekoppeltes Parametrisierungs-Diskretisierungsschema aus stückweise polinomialen Ansatz für die Steuerung und einen Runge-Kutta-Verfahren zur Integration der Zustands-Differentialgleichung. Die Konvergenzordnung der Lösung wird unter Regularitätsbedingung und Optimalitätsbedingung 2.Ordnung ermittelt. Außerdem wird eine Möglichkeit zur numerischen Berechnung der Gradienten über internen numerischen Differentiation erläutert. Schließlich werden einige numerischen Resultate gegeben und die Abhängigkeiten zur den theoretischen Konvergenzresultate diskutiert. / The focal point of this work is the investigation of a class of discretization methods for nonlinear optimal control problems governed by ordinary differential equations with control restrictions, as well as the implementation of some numerical experiments. The theoretical investigations are based on a coupledparameterization-discretization pattern, a piecewise linear parameterization representation of the control, and the application of a Runge Kutta method for the integration of the differential state equation. The rate of convergence of the solution is obtained with the help of regularity conditions and the second order optimality conditions. Furthermore, we also present in this paper a possibility of the numerical computation of the gradients via numerical differentiation. Finally some numerical results are given and their relationship to the theoretical convergence results are discussed.