Runge-Kutta型波形松弛方法的A-稳定  

A-STABILITY OF RUNGE-KUTTA TYPE OF WAVEFORM RELAXATION METHODS

作　　者：范振成 Fan Zhencheng(College of mathematic and data science,Minjiang University,Fuzhou 350108,China)

机构地区：[1]闽江学院数学与数据科学学院,福州350108

出　　处：《数值计算与计算机应用》2023年第3期327-336,共10页Journal on Numerical Methods and Computer Applications

基　　金：福建省自然科学基金(2021J011031);福建省发树慈善基金会资助研究专项(MFK23013)资助。

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,诸如线性多步法和Runge-Kutta(RK)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,学者们提出了波形松弛(WR)方法.多数情况下,这些微分方程组是刚性的,求解他们需要稳定性好的隐式方法,尤其需要A-稳定的方法.此外,RK方法是使用最广泛的常微分方程的数值方法.然而,迄今为止尚未发现RK型WR方法A-稳定的研究.本文研究了RK型WR方法的A-稳定性,获得了方法A-稳定的充分条件.常见A-稳定的RK方法有Gauss-Legendre方法、Radau ⅠA方法、Radau ⅡA方法、Lobatto ⅢA方法、Lobatto ⅢB方法、Lobatto ⅢC方法,而且并非RK方法A-稳定,相应的RK型WR方法也A-稳定.在一个假设下,本文所得结果说明当选择低阶Radau ⅠA方法,或Radau IIA方法,或Lobatto ⅢC方法为底层方法时,存在分裂方式使得RK型WR方法是A-稳定的.The high-dimensional differential and algebraic equations are common models of describing the running law of chips and electrical power systems,where the dimension of differential equations is too large to be solved effectively by using the classical numerical methods such as linear multistep methods and Runge-Kutta(RK)methods,ect.In order to solve such diferential equation,the waveform relaxation(WR)method is proposed.In most cases,the differential equations mentioned above are stiff.Hence the implicit methods with very good stability property are needed to solve effectively them,and especially an A-stable method is needed.In addition,the RK method is the numerical scheme of ordinary differential equations which is employed most extensively.However,so far the works on A-stability of RK type of WR methods are not found.In this paper,we investigate A-stability of RK type of WR methods,and obtain the suficient conditions of A-stability.The Gauss-Legendre method,Radau ⅠA method,Radau ⅡA method,Lobatto ⅢA method,Lobatto ⅢB method and Lobatto ⅢC method are A-stable Runge-Kutta methods used extensively.Under a common assumption,the obtained results show that there exist the splitting way such that the RK type of WR methods are A-stable when underlying methods are chosen as Radau ⅠA methods or Radau ⅡA methods or Lobatto ⅢC methods with low orders.

关 键 词：A-稳定 波形松弛方法 RUNGE-KUTTA方法 刚性方程 

分 类 号：O241.8[理学—计算数学]