CAIMS Prize Award

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Department of Computer Science, University of Toronto

Supervisor:  Wayne Enright

Thesis title:  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.