Part I: Folds and bifurcations in the solutions of semi-explicit differential-algebraic equations. Part II: The recursive projection method applied to differential-algebraic equations and incompressible fluid mechanics

Citation

Von Sosen, Harald (1994) Part I: Folds and bifurcations in the solutions of semi-explicit differential-algebraic equations. Part II: The recursive projection method applied to differential-algebraic equations and incompressible fluid mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JP6Q-ED69. https://resolver.caltech.edu/CaltechETD:etd-07282005-161146

Abstract

Part I: Folds and Bifurcations in the Solutions of Semi-Explicit Differential-Algebraic Equations

A general existence theory for the solutions of semi-explicit differential-algebraic 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 Differential-Algebraic 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 differential-algebraic equations (DAEs). This is accomplished by reformulating the DAE as an ODE. In particular, this approach applies to DAEs obtained by semi-discretization of the incompressible Navier-Stokes equations by use of the method of lines. The method is applied to compute unstable steady states of the flow between concentric rotating cylinders.

Thesis Files