Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations
Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations
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DOI：
10.1007/s002110050300
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发表时间：
1997-10
影响因子：
2.1
通讯作者：
M. Zennaro
中科院分区：
文献类型：
--
作者：
M. Zennaro
本文考虑一类时滞微分方程系统，其初始条件为.最近，Torelli [10]提出了耗散非线性微分方程数值方法的稳定性概念，即RN-稳定性，它是耗散常微分方程数值方法的BN-稳定性概念的直接推广。耗散性是指解和分别对应于不同的初始函数和，满足不等式y（t）-z（t）\|\leq \max_{x\leq t_0} \|\phi（x）-\psi（x）\|，\ t\geq t_0}$，并由适当的右端函数Lipschitz常数条件保证.一个数值方法被称为RN-稳定的，如果它保持这种收缩性质。在表明，稍微更严格的假设下的Lipschitz常数和延迟函数，solutionsandare这样，在本文中，我们证明RN稳定的连续Runge-Kutta方法也保持这种渐近稳定性。
We consider systems of delay differential equations (DDEs) of the formwith the initial condition. Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namelyRN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutionsandcorresponding to different initial functionsand, respectively, satisfy the inequality $\|y(t)-z(t)\|\leq \max_{x\leq t_0} \|\phi(x)-\psi(x)\|, \ t\geq t_{0}$, and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function. A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function, the solutionsandare such that, in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property.