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 $\theta$-methods with $\tfrac 12\leq \theta \leq 1$, the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.
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