Title page for ETD etd-08042002-140025

Type of Document

Dissertation

Author

Coffey, Todd Stirling,

Author's Email Address

todd@ecoffey.com

URN

etd-08042002-140025

Title

Temporal and Pseudo-Temporal Numerical Integration Methods

Degree

PhD

Graduate Program

Mathematics

Advisory Committee

Advisor Name	Title
C. S. Woodward	Committee Member
C. T. Kelley	Committee Member
D. S. McRae	Committee Member
M. Shearer	Committee Member
P. A. Gremaud	Committee Member

Keywords

interpolation
PTC
hyperbolic
elliptic
DAE
CFD

Date of Defense

2002-08-01

Availability

unrestricted

Abstract

COFFEY, TODD STIRLING. Temporal and Pseudo-Temporal Numerical Integration
Methods. (Under the direction of C. T. Kelley).

Numerical methods for integrating partial differential equations are used
in nearly every scientific field. In this dissertation we study two types
of numerical integration methods, transient methods and pseudo-transient
methods. Transient methods for partial differential equations look for time-accurate
solutions that explain the evolution of the equation (although a steady state
solution may evolve). Pseudo-transient methods look for steady-state solutions
of partial differential equations while paying attention to the transient
behavior to aid in stability. In contrast, root-finding methods, e.g. line-search
methods, look only for a steady-state solution often not paying attention
at all to the transient behavior of the problem.

Pseudo-transient continuation is a method for solving steady state solutions
of partial differential equations, and is used when traditional line-search
methods fail to converge or converge to non-physical solutions. The method
is a hybrid between implicit Euler and Newton's method where variable step-sizes
are used to transfer from one method to the other. We demonstrate the performance
of pseudo-transient continuation both numerically and theoretically on a variety
of problems. We extend the global convergence theory, which currently covers
a class of ordinary differential equations, to include a class of semi-explicit
index-1 differential-algebraic equations.

We also studied CVode, a transient code for solving nonlinear
partial differential equations. In this work, we explain how CVode
was extended to allow for a two-grid nonlinear solver. The two-grid solver
coarsens a given mesh and solves the nonlinear problem on the coarse mesh,
which is then moved back to the fine mesh for refining. This approach can
be less expensive than computing the full nonlinear solution on the fine
mesh. We explore some of the theoretical and computational issues involved
in implementing this method for a radiative transfer problem as might be
seen in stellar fusion.

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