非线性刚性变延迟微分方程单支方法的数值稳定性

现有文献中对于非线性延迟微分方程渐近稳定性及其数值方法的稳定性研究大都局限于常延迟的情形,例如可参见匡蛟勋[1-3],黄乘明[4],Torelli[5]等人的大量工作.1994年A.Iserles[6] 首次研究了比例延迟微分方程数值方法的线性稳定性,随后有相当多的文献对比例延迟微分方程的各种数值方法的线性稳定性进行了讨论.1997年Zennaro[7]首次研究了非线性刚性变延迟微分方程的渐近稳定性,但该文中对于延迟量的限制十分苛刻,同时该文也首次研究了非线性刚性变延迟微分方程Runge-Kutta方法的非线性稳定性. 本文目的是试图在上述基础上进一步研究非线性刚性变延迟微分方程的渐近稳定性及其数值方法的稳定性.首先在第二节我们给出了非线性刚性变延迟微分方程模型问题（2.1）渐

THE NUMERICAL STABILITY OF ONE-LEG METHODS FOR NONLINEAR STIFF DELAY DIFFERENTIAL EQUATIONS WITH A VARIABLE DELAY

In this paper, we discuss the asymptotic stability of nonlinear Delay Differential Equations(DDEs) with a variable delay and the numerical stability of one-leg methods when applied to such equations. At first, we give a sufficient condition for the aforementioned equations to be asymptotic stable, which is simpler and more effective than that presented by Zennaro in 1997. For one-leg methods applied to nonlinear DDEs with a variable delay, we introduce a series of new stability concepts, such as GR(v)-stability, GAR(v)-stability and weak GAR(v)-stability, where v > 0 is a given constant. Afterwards, for one-leg methods with linear interpolation, we prove that A-stability implies GR(2~(1/2)/2)-stability and weak GAR(2~(1/2)/2)-stability, and that strong A-stability implies GAR(2~(1/2)/2)-stability for delay differential equations with a variable delay. Several numerical tests listed at the end of this paper to confirm the above theoretical results.