Highly-Stable Multistep Methods for Retarded Differential Equations.

摘要

A linear multistep method (rho,sigma) is defined to be D(A sub 0)-stable if when it is applied to the delay differential equation dy/dt(t) =mu y(t-tau) the approximate solution (y sub h)(t sub n) nears 0as n nears infinity for all mu epsilon(o,pi/2tau) and all stepsized h of the form h =tau/m, m a positive integer. General properties of D(A sub 0)-stable methods are derived. These properties are similar to the properties of A-stable and A(alpha)-stable methods,for example,it is proved that a k-step D(A sub 0)-stable method or order k must be implicit. As an application it is shown that the trapezoidal method is D(A sub 0)-stable. Finally,the condition that h =tau/m is dropped and the resulting methods,which the author calls GD(A sub 0)-stable methods,are studied. (Author)