| The first part of the paper studies constant coe?cient linear scalar delaydi?erential equations , and proves both implicit mid-point method and trape-zoidal rule is B-stable and conditionally contractive. Furthermore, the sameconclusion can be similarly promoted for a class of variable coe?cient semi-linearscalar Volterra functional di?erential equations. The two examples Bellen men-tioned in the chapter 1 of the book $Numerical methods for delay di?erentialequations |indicated that 2 stage Lobatto IIIC Runge-Kutta method is a goodmethod while neither mid-point method nor trapezoidal rule is B-stable. Theconclusion is a little defective. In fact,the two methods above are B-stable andconditionally contractive for solving the two examples given in the first chapterof Bellen's book, only if the step size of the examples is decreased appropriately.What's more, the result of solving the problem in example 1 indicates the practi-cal calculation e?ciency of both implicit mid-point method and trapezoidal ruleis much higher than 2 stage Lobatto IIIC Runge-Kutta method (Lobatto2).The second part of this paper studies the general nonlinear sit? DDEs, andcompares the 2 stage Runge-Kutta methods such as Lobatto IIIC ,Radau IIA andGauss.Both theoretical analysis and numerical experimentations indicate that theabove three methods cost almost identical time in every integral step, but thecomputational accuracy of the latter two methods is much higher than Lobatto2.It is seen from the above that Lobatto2 is not worth to be recommended. Wesuggest using the Radau IIA Runge-Kutta method in practice.When the sti?nessof functional part is not sharp, the Gauss Runge-Kutta method can be consideredfirstly.Among international literatures, few paper give similar results and compar-isons about 2 stage Lobatto IIIC Runge-Kutta method, mid-point method andtrapezoidal rule. So this paper achieves innovation to some extent and can providesome reference for practice calculation. |
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