一类特殊的块方法

A Special Class of Block Methods

在求解常微分方程和微分代数方程中,块方法是一种有效的方法。这类方法是单步的,且其数值精度不受数值稳定性的约束,因而比线性多步法更适应于求解刚性微分方程或者高指标微分代数方程。但是,以往的块方法因为其巨大的计算工作量而未被广泛使用。本文研究了一类块方法,使其构成矩阵只含有一个重特征值,因而在隐式速代时,计算量大致上与线性多步法相当。本文讨论了该特征值与Lagurre多项式的关系,从而建立了这类块方法的构成公式,数值试验证明了理论上得到的计算量的估计。

In numerically solving ordinary differential equations and differential-algebraic equations block methods are efficient. They are one-step methods. Their convergent order, on the other hand ( is not restricted by their numerical stability. So, they are suitable in solving Stiff ODE and DAE with high indices. However, They were not widely accepted because of their huge amount of computation. In this paper, a class of new block methods are constructed. There is only one k -fold eigenvalue In the coefficlant matrix. Therefore) the working amount when Newton-Raph-son iteration is carried out is equivalent to that of the linear multistep method. The relation between the eigenvalue of the coefficient matrix and the zeros of Lagurre polynomial is also discussed, so we can construct this class of methods easily. Numerical tests show that the theoretic estimate of the working amount is reliable.