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Interpolating Runge-Kutta methods for vanishing delay differential equations. (English) Zbl 0843.65052
The authors describe a new method which modifies continuous Runge-Kutta (RK) codes so as to be applicable to the approximation of the solution of differential delay problems which have a delay which vanishes. For a delay function which vanishes during a step \([t_n, t_{n+ 1}]\) of the calculation the new algorithm uses values from the preceding step \([t_{n- 1}, t_n]\) to generate interpolants needed to construct the stage values for the \(n\)th step. The order of accuracy of the modified method is designed to be equal to that of the original method.
The authors have written codes DDRK6N, DDVSS6 based on the new algorithm which are sixth-order RK codes from the Verner class [c.f. J. H. Verner, SIAM J. Numer. Anal. 30, No. 5, 1446-1466 (1993; Zbl 0787.65047)]. Results obtained by application of these codes to three test problems indicate that the new codes are more accurate than existing codes.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34K05 General theory of functional-differential equations
Software:
DDRK6N; DDVSS6; dverk; SYSDEL
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References:
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