Strong contractivity properties of numerical methods for ordinary and delay differential equations
Strong contractivity properties of numerical methods for ordinary and delay differential equations
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DOI：
10.1016/0168-9274(92)90025-9
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发表时间：
1992-04
影响因子：
2.8
通讯作者：
A. Bellen;M. Zennaro
中科院分区：
文献类型：
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作者：
A. Bellen;M. Zennaro
在过去的 20 年里，DDE 求解器的各种稳定性测试问题都得到了考虑。这些主要是用于 ODE 的概括。对于最简单的自治情况 y′(t)= λy (t)+ μy (t− τ)，引入了 P 稳定性和 GP 稳定性的概念，并且已经为线性多步方法和一步龙格-库塔方法找到了大量结果。更困难的是非自治线性检验方程 y′(t)= λ (t) y (t)+ μ (t) y (t− τ) 和一般非线性情况 y′(t)= f (t, y (t), y (t− τ)) 的情况，这产生了 PN-、GPN-、RN-和 GRN-稳定性的概念。特别是，为了研究 ODE 求解器的 PN 和 GPN 稳定性概念，基于具有强制项 y′(t)= λ (t) y (t)+ f (t) 的测试方程，最近发现了称为 AN f 稳定性的概念。本文进一步探讨了这一点，并引入了 A f 稳定性和 BN f 稳定性的概念，它们分别基于检验方程 y′(t)= λy (t)+ f (t) 和 y′(t)= f (t, y (t), u (t))。所有这些 ODE 和 DDE 稳定性概念之间都建立了一些关系。彻底检查了 2 阶龙格-库塔方法类的情况。
In the last 20 years various stability test problems for DDE solvers have been considered. These are mainly generalizations of those used for ODEs. For the simplest autonomous case y′(t)= λy (t)+ μy (t− τ), the concepts of P-and GP-stability were introduced and a significant number of results have already been found for both classes of linear multistep methods and one-step Runge-Kutta methods. More difficult is the situation for the non-autonomous linear test equation y′(t)= λ (t) y (t)+ μ (t) y (t− τ) and for the general nonlinear case y′(t)= f (t, y (t), y (t− τ)), which gave rise to the concepts of PN-, GPN-, RN-and GRN-stability. In particular, in order to study PN-and GPN-stability notion for ODE solvers based on the test equation with forcing term y′(t)= λ (t) y (t)+ ƒ (t), which is called AN ƒ-stability, has recently been found. This paper pursues this further and introduces the concepts of A ƒ-stability and BN ƒ-stability, which are based on the test equations y′(t)= λy (t)+ ƒ (t) and y′(t)= ƒ (t, y (t), u (t)), respectively. Some relationships are established among all these concepts of stability for ODEs and DDEs. The situation for the class of Runge–Kutta methods up to order 2 is thoroughly examined.