非线性脉冲微分方程的Runge-Kutta方法的稳定性分析

考虑了一般的非线性脉冲微分方程,对该方程进行了解析解和数值解的稳定性分析.在不受脉冲影响的原方程满足单边Lipschitz条件,及脉冲项满足相应的Lipschitz条件的情况下,给出了一个容易判别的解析解渐近稳定的充分条件.把脉冲点作为节点,定义了一个收敛的变步长的Runge-Kutta方法.并且证明了如果一个方法是代数稳定的,则该方法的数值解保持解析解的渐近稳定性.

Stability of Runge-Kutta methods in the numerical solution of nonlinear impulsive differential equations

The stability analysis of the analytic and numerical solutions of the general nonlinear im-pulsive differential equation is considered. Under the conditions that the system without impulse effect satisfies the one-side Lipschitz condition, and the impulsive terms satisfy the corresponding Lipschitz conditions, a sufficient condition which can be easily checked the stability of the analytic solution is ob-tained. Furthermore, by taking the instants of the impulse effects as the nodes, a convergent variable stepsize Runge-Kutta method is defined. Moreover, if a method is algebraically stable, then the numeri-cal solutions of this method can preserve the stability property of the analytic ones.