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CAMS/SCMA Doctoral Dissertation Award 85
Winner and Abstract
1985
Wendy Seward (Wayne Enright),
Department of Computer Science,
University of Toronto
Defect and Local Error Control in Codes for Solving Stiff
Initial-Value Problems
The aim of this thesis is to motivate and develop error control strategies
that lead to a uniform interpretation of the user's requested accuracy in
codes for solving stiff initial-value problems. Tolerance proportionality
is identified as a minimum requirement for such a uniform interpretation
and results are quoted that show that a code must control the local error
per unit step in order to achieve tolerance proportionality. A simple
modification of the existing local-error-per-step error control strategy
used by two popular stiff solvers leads to some improvement in tolerance
proportionality. As an alternative, local extrapolation based on an
estimate of the error in the corrected value is used to achieve a
generalized local error per unit step error control. This approach improves
tolerance proportionality for one of the codes but not for the other. In
nonstiff solvers, local extrapolation is usually done based on an estimate
of the error in the predicted value. This approach is shown to be easy to
implement and an effective means of improving tolerance proportionality for
both codes.
Tolerance proportionality and defect proportionality are closely related,
and the behaviour of the defect in some continuous approximate solutions
naturally associated with the existing codes is investigated. It is shown
that even when the codes exhibit reasonable tolerance proportionality at
each discrete step, these obvious continuous approximations do not
necessarily exhibit defect proportionality.
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