一类线性隐式A稳定的单步法

1.引言 对于Stiff方程组初值问题的数值解法,Dahlquist在1]中引进了 A稳定的概念,并且证明了显式的线性多步法(包括显式的Runge-Kutta方法)不可能是A稳定的.现在已经有许许多多隐式A稳定或Stiff稳定的方法,但绝大多数在数值解的过程中必须解由于隐式方法所产生的非线性方程组,而非线性方程组的求解过程往往又要采用Newton-Raphson迭代方法,因此需要计算方程y’=f(x,y)的右函数f(x,y)的Jacobi矩阵以及与此有关的逆矩阵.本文的主要思想是:既然在数值解过程中要计算f(x,y)的Jacobi矩阵,那么不妨在数值公式中明显的出现f(x,y)的一阶偏导数.我们将A稳定公式

A CLASS OF LINEARLY IMPLICIT A-STABLE ONE-STEP METHODS

For Stiff initial-value problems including a system of s coupled, non-linear,first-order ordinary differential equations, a class of linearly implicit A-stable one-step methodsis developed. These methods may be derived from suitably modified A-stable one-step methodscombined with the idea of the derivation of classical Runge-Kutta methods. We have constructedk-order linearly implicit A-stable one-step methods, k = 1, 2, 3, 4. Applying such technique to Gear's method, One may obtain a class of linearly implicitstiffly stable multistep methods.