Modern numerical ordinary differential equation initial-value problem
(ODE-IVP) solvers compute a piecewise polynomial approximate solution
to the mathematical problem. Evaluating the mathematical problem at
this approximate solution defines the defect. Corless and Corliss
proposed rigorous defect control of numerical ODE-IVP.
This thesis automates rigorous defect control for explicit,
first-order, nonlinear ODE-IVP. Defect control is residual-based
backward error analysis for ODE, a special case of Wilkinson's
backward error analysis. This thesis describes a complete software
implementation of the Corless and Corliss algorithm and extensive
numerical studies. Basic time-stepping software is adapted to defect
control and implemented.
Advances in software developed for validated computing applications
and advances in programming languages supporting operator overloading
enable the computation of a tight rigorous enclosure of the defect
evaluated at the approximate solution with Taylor models. Rigorously
bounding a norm of the defect, the Corless and Corliss algorithm
controls to mathematical certainty the norm of the defect to be less
than a user specified tolerance over the integration interval. The
validated computing software used in this thesis happens to compute
a rigorous supremum norm.
The defect of an approximate solution to the mathematical problem
is associated with a new problem, the perturbed reference problem.
This approximate solution is often the product of a numerical procedure.
Nonetheless, it solves exactly the new problem including all errors.
Defect control accepts the approximate solution whenever the sup-norm
of the defect is less than a user specified tolerance. A user must be
satisfied that the new problem is an acceptable model. / Thesis / Master of Science (MSc) / Many processes in our daily lives evolve in time, even the weather.
Scientists want to predict the future makeup of the process. To do
so they build models to model physical reality.
Scientists design algorithms to solve these models, and the algorithm
implemented in this project was designed over 25 years ago. Recent
advances in mathematics and software enabled this algorithm to be
implemented.
Scientific software implements mathematical algorithms, and
sometimes there is more than one software solution to apply to the
model. The software tools developed in this project enable
scientists to objectively compare solution techniques.
There are two forces at play; models and software solutions.
This project build software to automate the construction of the
exact solution of a nearby model. That's cool.
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/22733 |
| Date | 10 1900 |
| Creators | Ernsthausen, John+ |
| Contributors | Nedialkov, Nedialko, Computational Engineering and Science |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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