Runge-Kutta type methods for differential-algebraic equations in mechanics
Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and e...
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ndltd-uiowa.edu-oai-ir.uiowa.edu-etd-24672019-10-13T04:52:39Z Runge-Kutta type methods for differential-algebraic equations in mechanics Small, Scott Joseph Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered. 2011-05-01T07:00:00Z dissertation application/pdf https://ir.uiowa.edu/etd/1082 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=2467&context=etd Copyright 2011 Scott Joseph Small Theses and Dissertations eng University of IowaJay, Laurent O. Differential-Algebraic Equations Gauss-Lobatto Coefficients Holonomic Constraints Lagrangian Systems Nonholonomic Constraints Runge-Kutta Methods Applied Mathematics |
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Differential-Algebraic Equations Gauss-Lobatto Coefficients Holonomic Constraints Lagrangian Systems Nonholonomic Constraints Runge-Kutta Methods Applied Mathematics |
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Differential-Algebraic Equations Gauss-Lobatto Coefficients Holonomic Constraints Lagrangian Systems Nonholonomic Constraints Runge-Kutta Methods Applied Mathematics Small, Scott Joseph Runge-Kutta type methods for differential-algebraic equations in mechanics |
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Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3.
Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods.
When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s.
Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators.
This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered. |
author2 |
Jay, Laurent O. |
author_facet |
Jay, Laurent O. Small, Scott Joseph |
author |
Small, Scott Joseph |
author_sort |
Small, Scott Joseph |
title |
Runge-Kutta type methods for differential-algebraic equations in mechanics |
title_short |
Runge-Kutta type methods for differential-algebraic equations in mechanics |
title_full |
Runge-Kutta type methods for differential-algebraic equations in mechanics |
title_fullStr |
Runge-Kutta type methods for differential-algebraic equations in mechanics |
title_full_unstemmed |
Runge-Kutta type methods for differential-algebraic equations in mechanics |
title_sort |
runge-kutta type methods for differential-algebraic equations in mechanics |
publisher |
University of Iowa |
publishDate |
2011 |
url |
https://ir.uiowa.edu/etd/1082 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=2467&context=etd |
work_keys_str_mv |
AT smallscottjoseph rungekuttatypemethodsfordifferentialalgebraicequationsinmechanics |
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1719265086929895424 |