Adaptive Integration of Delay Differential Equations
We consider the numerical solution of nonstiff delay differential equations by means of a variable stepsize continuous Runge-Kutta method, the advancing method, of discrete order p and uniform order q ≤ p. As in the well-known case of ordinary differential equations, the stepsize control mechanism is based on the use of a second method, the errore-stimating method, of order p’ ≠ p, used to adapt the curret stepsize in order that the local error fits a user-supplied tolerance TOL. We detect the minimal uniform order q needed for the correct performance of both the advancing and the error-estimating methods. Arter stating the relationships among global and local errors, we also discuss the effectiveness of the stepsize control mechanism in connection to the possible use of an additional continuous error-estimating method, which is used to monitor the unifonn local error. A complete and detailed description of both the advancing method of unifonn order 4 and the disc rete error-estimating method of order 5 is given so as to enable the interested reader to implement his own code. A unifonn error-estimating mechanism is also given which is recommendable for the reliability of the overall procedure. Finally. numerical experiments with a constructed equation are carried out aimed to test and check the resu lts provided by the theory.
KeywordsLocal Error Delay Differential Equation Global Error Delay Differential EQuations Variable Stepsize
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