Citation
Von Sosen, Harald (1994) Part I: Folds and bifurcations in the solutions of semiexplicit differentialalgebraic equations. Part II: The recursive projection method applied to differentialalgebraic equations and incompressible fluid mechanics. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd07282005161146
Abstract
Part I: Folds and Bifurcations in the Solutions of SemiExplicit DifferentialAlgebraic Equations
A general existence theory for the solutions of semiexplicit differentialalgebraic equations (DAEs) is given. Theorems on the form and number of solutions in a neighborhood of an initial value are presented. A set of bifurcation equations is derived, from which the tangents of these solutions can be computed. The phenomena of folds and bifurcation are studied. It is shown that solutions near fold points and pitchfork bifurcation points can be represented smoothly if an appropriate parametrization is introduced. Moreover, it is shown that the complex analytic extension of a real DAE often has complex solutions near a real initial value, and existence theorems on these complex solutions are given. Examples from electrical engineering are presented in support of the theory. Methods for adapting existing numerical DAE solvers to handle fold and bifurcation points are introduced. These methods are tested on a nonlinear electric circuit problem.
Part II: The Recursive Projection Method Applied to DifferentialAlgebraic Equations and Incompressible Fluid Mechanics
The Recursive Projection Method (RPM) was originally invented by Schroff and Keller for the stabilization of unstable fixed point iterations. A direct application of RPM lies in the computation of unstable steady states of nonlinear ordinary differential equations (ODEs) via time integration. Here, the method is generalized to handle algebraic constraints so that it can be applied to certain differentialalgebraic equations (DAEs). This is accomplished by reformulating the DAE as an ODE. In particular, this approach applies to DAEs obtained by semidiscretization of the incompressible NavierStokes equations by use of the method of lines. The method is applied to compute unstable steady states of the flow between concentric rotating cylinders.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied And Computational Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  5 May 1994 
NonCaltech Author Email:  Harald.Vonsosen (AT) synopsys.com 
Record Number:  CaltechETD:etd07282005161146 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd07282005161146 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2986 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  28 Jul 2005 
Last Modified:  26 Dec 2012 02:56 
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