Controlling the defect in existing variable-order Adams codes for initial-value problems

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Abstract
Variable-order, variable-step multistep methods based on Adams formulas have proved very effective in the numerical solution of nonstlff systems of ordinary differential equations. The user specifies an accuracy parameter and the method attempts to produce a solution consistent with this accuracy requirement. The relationship between the global error in the numerical solution and the prescribed accuracy requirement is problem dependent and very difficult to quantify precisely. On the other hand, most variable-order Adams methods provide the user with a piecewise polynomial approximation to the solution and use this to provide intermediate solutions at specified output points. Since the pmcewlse polynomial is dlfferentlable at all but a fimte number of points, one can define the defect of the numerical solution as the amount by which the plecewlse polynomial fails to satisfy the differential equation The maximum magnitude of the defect can then be used as a measure of the accuracy of the solution and, since the defect can be shown to be directly related to the local error, it provides a measure of accuracy that is relatively insensitive to the specific problem. In this study, the underlying piecewise polynomial approximates associated with two of the popular Adams codes are analyzed and assessed, and it is shown how the magnitude of the defect Is related to the prescribed accuracy parameter That current methods can keep the maximum magnitude of the defect bounded by a small multiple of the tolerance is also shown Numerical results that verify the analysis on a variety of problems are presented.

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