An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as additively split when it is comprised of the sum of two or more contributing factors. For instance, the right-hand sides of initial-value problems derived from advection-diffusion-reaction equations are comprised of the sum of terms emanating from three distinct physical processes: advection, diffusion, and reaction. In some cases, solutions to initial-value problems can be calculated analytically, but when an analytic solution is unknown or nonexistent, methods of numerical integration are used to calculate solutions. The runtime performance of numerical methods is problem dependent; therefore, one must choose an appropriate numerical method to achieve favourable performance, according to characteristics of the problem. Additive methods of numerical integration apply distinct methods to the distinct contributing factors of an additively split problem. Treating the contributing factors with methods that are known to perform well on them individually has the potential to yield an additive method that outperforms single methods applied to the entire (unsplit) problem. Splittings of the right-hand side can be physics-based, i.e., based on physical characteristics of the problem, such as advection, diffusion, or reaction terms. Splittings can also be based on linearization, called Jacobian splitting in this thesis, where the linearized part of the problem is treated with one method and the rest of the problem is treated with another. A comparison of these splitting techniques is performed by applying a set of additive methods to a test suite of problems. Many common non-additive methods are also included to serve as a performance baseline. To perform this numerical study, a problem-solving environment was developed to evaluate permutations of problems, methods, and their associated parameters. The test suite is comprised of several distinct advection-diffusion-reaction equations that have been chosen to represent a wide range of common problem characteristics. When solving split problems in the test suite, it is found that additive Runge–Kutta methods of orders three, four, and five using Jacobian splitting generally outperform those same methods using physics-based splitting. These results provide evidence that Jacobian splitting is an effective approach when solving such initial-value problems in practice.

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In this work, we present an amplitude-shape method for solving evolution problems described by partial differential equations. The method is capable of recognizing the special structure of many evolution problems. In particular, the stiff system of ordinary differential equations resulting from the semi-discretization of partial differential equations is considered. The method involves transforming the system so that only a few equations are stiff and the majority of the equations remain non-stiff. The system is treated with a mixed explicit-implicit scheme with a built-in error control mechanism. This approach proved to be very effective for the solution of stiff systems of equations describing spatially dependent chemical kinetics. / Thesis (Ph.D.)-University of Natal, 1997.

This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under Hölder-type source-wise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.

Zugl.: Bayreuth, Univ., Diss., 2006 u.d.T.: Korch. Matthias: Effiziente Implementierung eingebetteter Runge-Kutta-Verfahren durch Ausnutzung der Speicherzugriffslokalität / Hergestellt on demand

Universiẗat, Diss., 2000--Göttingen.

This thesis considers the application of the type II third order WENO finite volume reconstruction for unstructured tetrahedral meshes proposed by Zhang and Shu in (CCP, 2009) and the third order multirate Runge-Kutta time-stepping to the solution of Maxwell's equations. The dependance of accuracy of the third order WENO scheme on the small parameter in the definition of non-linear weights is studied in detail for one-dimensional uniform meshes and numerical results confirming the theoretical analysis are presented for the linear advection equation. This analysis is found to be crucial in the design of the efficient three-dimensional WENO scheme, full details of which are presented. Several multirate Runge-Kutta (MRK) schemes which advance the solution with local time-steps assigned to different multirate groups are studied. Analysis of accuracy of three different MRK approaches for linear problems based on classic order-conditions is presented. The most flexible and efficient multirate schemes based on works by Tang and Warnecke (JCM, 2006) and Liu, Li and Hu (JCP, 2010) are implemented in three-dimensional finite volume time-domain (FVTD) method. The main characteristics of chosen MRK schemes are flexibility in defining the time-step ratios between multirate groups and consistency of the scheme. Various approaches to partition the three-dimensional computational domain into multirate groups to maximize the achievable speedup are discussed. Numerical experiments with three-dimensional electromagnetic problems are presented to validate the performance of the proposed FVTD method. Three-dimensional results agree with theoretical and numerical accuracy analysis performed for the one-dimensional case. The proposed implementation of multirate schemes demonstrates greater speedup than previously reported in literature. / February 2016

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-05-22Bitstream added on 2014-06-13T20:35:12Z : No. of bitstreams: 1 campos_jr_me_sjrp.pdf: 474631 bytes, checksum: 9a9f4df9bf2898f15cba64a064eec09b (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Métodos de Runge-Kutta para problemas de controle ótimo contínuo são estudados seguindo os trabalhos de Hager [11], [15] e [17]. O problema de controle ótimo é discretizado transformando-se num problema de programação matemática. Um estudo sobre as condições necessárias de otimalidade para a solução do problema e conexões com o problema adjunto é realizado para obtenção das condições de ordem na discretização. Estuda-se também a convergência da solução do problema discretizado para a solução ótima do problema contínuo (ver Hager [17]). Nesta análise Hager obtêm uma cota para o erro entre a solução numérica e a solução contínua o qual depende do tamanho do passo. Por fim, o trabalho apresenta alguns exemplos com o intuito de ilustrar a teoria apresentada. / Runge-Kutta methods for continuous optimal control problems are studied following the papers of Hager [11], [15] and [17]. The control problem is discretized and transformed into a mathematical programming problem. A study about necessary conditions of optimality for the solution of the problem and connections with an adjoint problem are done to provide order conditions for the method of discretization. It is also studied the convergence of the optimal solution of the discrete problem for the solution of the continuous time control problem (see Hager [17]). In this convergence analysis Hager obtains an error bound comparing the numerical and the continuous solution. The error bound is dependent of the size of the step of the method. Finally, some examples are presented aiming at illustrating the discussed theory.

Orientador: Suzete Maria Silva Afonso / Banca: Ligia Laís Fêmina / Banca: Marta Cilene Gadotti / Resumo: O propósito deste trabalho é explorar os métodos numéricos de passo único, denominados métodos de Euler e Runge - Kutta, e os métodos de passos múltiplos, denominados métodos de Adams-Bashforth e Adams-Moulton, para encontrar soluções aproximadas de problemas de valor inicial para equações diferenciais ordinárias de primeira ordem / Abstract: The goal of this work is to explore the one-step numerical methods, called methods of Euler and Runge - Kutta, and the multistep numerical methods, called methods of Adams-Bashforth and Adams Moulton, for finding approximate solutions of initial value problems for first-order ordinary differential equations / Mestre

As aplicações da mecânica vibratória vêm crescendo significativamente na análise de sistemas de suspensões e estruturas de veículos, dentre outras. Desta forma, o presente trabalho desenvolve técnicas para a simulação e o controle de uma suspensão de automóvel utilizando modelos dinâmicos com um, dois e três graus de liberdade. Na obtenção das equações do movimento para o sistema massa-mola-amortecedor, o modelo matemático utilizado tem como base a equação de Lagrange e a segunda lei de Newton, com condições iniciais apropriadas. A solução numérica destas equações é obtida através do método de Runge-Kutta de 4ª ordem, utilizando o software MATLAB. Para controlar as vibrações do sistema utilizou-se três métodos diferentes de controle: clássico, LQR e alocação de pólos. O sistema assim obtido satisfaz as condições de estabilidade e de desempenho e é factível para aplicações práticas, pois os resultados obtidos comparam adequadamente com dados analíticos, numéricos ou experimentais encontrados na literatura, indicando que técnicas de controle como o clássico podem ser simples e eficientes.