Control of Interpolatory Error in Retarded Differential Equations

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Abstract
The numerical solution of several large subclasses of functional differential equations requires interpolation over the discrete history of the solution. For example, interpolation is required to evaluate retarded arguments in FDEs with lags or perform quadrature in integrodifferential equations. The interpolatory process can be severely hampered by naturally occurring derivative discontinuities at possibly unknown points. In this work the class of retarded ordinary differential equations (RODEs) with state-dependent lags is considered. For such equations the propagation of derivative discontinuities has been completely characterized as the zeros of odd multiplicity of certain nonlinear equations involving the solution. It has been shown elsewhere that under liberal hypotheses these propagated zeros can be successively calculated to an (order of) accuracy that will preserve high-order convergence whenever the calculated (approximate) zeros are included in the integration mesh. This result depends on a key relation between the order of the discontinuous derivative and the multiplicity of the corresponding jump-point zero. Unfortunately, in vector equations the bookkeeping and overhead logic involved in solving these nonlinear equations is at best onerous. Another approach is to employ an automatic step-selection algorithm and to depend on the error control process to sense discontinuities and thereby use sufficiently small steps in the neighborhood of discontinuities to prevent large degradation in accuracy. It was conjectured elsewhere that this process of "ignoring the jump points" would not prevent a previously reported algorithm from obtaining the solution to the requested accuracy. In this work it is shown that the conjecture is dramatically false. The issue of derivative discontinuities as discussed is not peculiar to retarded differential equations. With proper interpretation these results also apply to ordinary differential equations (ODEs). It is shown that the detection of derivative discontinuities is not always reliable. Two methods are analyzed: the Merson method and the more popular Runge-Kutta Fehlberg (4, 5) formulas. It should be emphasized that the intent of using two methods is to illustrate that the conclusions are not peculiar to just one method. There is no intent to compare their relative performance in this discontinuous setting. Finally, an approach (also applicable to ODEs) is proposed that does not require the user to keep track of the potential cascade of jump points that occurs in RODEs or to be directly involved in setting up and/or solving the nonlinear jump-point equations. An error-control algorithm that estimates the. effects of a discontinuity through interpolation is suggested. Results are given using these modifications.

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