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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (1): 110-122.doi: 10.3969/j.issn.1005-3085.2023.01.008

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基于$\theta$方法的波形松弛方法的 A 稳定

范振成   

  1. 闽江学院数学与数据科学学院，福州 350108
  • 出版日期:2023-02-15 发布日期:2023-04-11
  • 基金资助:
    福建省自然科学基金(2021J011031).

A-stability of Waveform Relaxation Methods Based on $\theta$-methods

FAN Zhencheng   

  1. College of Mathematics and Data Science, Minjiang University, Fuzhou 350108
  • Online:2023-02-15 Published:2023-04-11
  • Supported by:
    The Natural Science Foundation of Fujian Province (2021J011031).

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摘要：

描述芯片或电力系统运行规律的常用数学模型是高维微分代数方程组，其中的微分方程组太大，线性多步法和Runge-Kutta法等经典数值方法均不能有效求解。为求解这些微分方程组，借鉴常微分方程经典数值方法的A稳定定义，提出了波形松弛方法A稳定(强A稳定)，给出了基于$\theta$方法的波形松弛方法A稳定(强A稳定)和非A稳定的条件，以及几个支持理论结果的数值算例。研究结果表明WR方法并非天然继承底层方法的A稳定性，为使波形松弛方法A稳定，需要使用A稳定的底层方法和适当的分裂函数，这为刚性方程WR方法的构造奠定了理论基础。此外，借鉴经典数值方法的B稳定定义，提出了波形松弛方法的B稳定(强B稳定)，给出了波形松弛方法强B稳定的条件。

关键词: 波形松弛方法, A稳定, B稳定, $\theta$方法, 刚性问题

Abstract:

The models describing the chip and electric systems are usually differential-algebraic equations of high dimension, and the dimension of the equations is too large to be solved effectively by the classical numerical methods such as linear multistep methods and Runge-Kutta methods. To solve these equations, by referencing the A-stability definition of the classical numerical methods of ordinary differential equations, A-stability (strong A-stability) is proposed for waveform relaxation (WR) methods, and the conditions of A-stability (strong A-stability) and non-A-stability and several numerical examples of supporting theoretical results are presented. The obtained results show that WR methods cannot inherit naturally A-stability of underlying numerical methods and one need use A-stable underlying numerical methods and suitable splitting functions for A-stability of WR methods. All these lay a theoretical foundation for constructing the WR methods of stiff systems. Furthermore, B-stability (strong B-stability) of WR methods is proposed by referencing the B-stability definition of the classical numerical methods, and the conditions of the strong B-stability are given.

Key words: waveform relaxation methods, A-stability, B-stability, $\theta$-methods, stiff problems

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