The stability of modified Runge-Kutta methods for the pantograph equation

Abstract

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the θ \theta -methods with 1 2 ≤ θ ≤ 1 \tfrac 12\leq \theta \leq 1 , the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.