多导块方法的稳定性分析
The Numerical Stabilities of Multiderivative Block Method
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摘要: 文[1]中提出了一种使用高阶导数的块隐式单步法,并在文末留下一个问题:即对给定的方法中使用的最高阶导数阶数l≥1,为使该方法是A-稳定的,块的大小k应满足什么条件?本文将彻底地解答这个问题.首先,我们给出稳定函数ξk(h)=P(h)/Q(h)中多项式P(h)及Q(h)的系数的显式表达式,并证明P(-h)=Q(h);另外,我们使用计算机符号运算及对角Pade'逼近公式,对任意的l≥1,给出了为使方法A-稳定时块的大小k应满足的条件.Abstract: In [1],a class of multiderivative block methods(MDBM) was studied for the numerical solutions of stiff ordinary differential equations.This paper is aimed at solving the problem proposed in [1] that what conditions should be fulfilled for MDBMs in order to guarantee the A-stabilities.The explicit expressions of the polynomials P(h) and Q(h) in the stability functions ξk(h)=P(h)/Q(h)are given.Furthermore,we prove P(-h)=Q(h).With the aid of symbolic computations and the expressions of diagonal Fade approximations,we obtained the biggest block size k of the A-stable MDBM for any given l(the order of the highest derivatives used in MDBM,l≥1)
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Key words:
- multiderivative block methods /
- A-stability /
- block size
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[1] 匡蛟勋,带有高阶导数的块隐式单步法,高校计算数学学报,9(1)(1987),15-23. [2] Watts,H.A.and L.E.Shampine:A-stable block implicit one-step methods,BIT,12(1972),252-266. [3] Birkhoff,G.and R.S.Varga,Discretization errors for well-set Cauchy problems(Ⅰ),J.Math,and Phys,4(1965),1-23. -
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