具极小化局部截断误差的Runge-Kutta方法

RUNGE-KUTTA METHODS WITH MINIMIZING THE LOCAL TRUNCATION ERROR

使用本文提出的既不增加函数f(x,y)求值的个数计算又不要求f(x,y)及(?)^(i+j)f/(?)x^i(?)y^j 为有界的方法,便建立了具极小化局部截断误差的二级二阶直到四级四阶的Runge-Kutta 公式.这些公式均可用于求解非线性一阶常微分方程组,且是对Lotkin(1951)、Ralston(1962)、Merson(1975)、Scraton(1964)、England(1969)的结果的一种改善和推广.此外,当常微分方程组退化成一个方程时,Lotkin(1951)和Ralston(1962)的若于结果就是本文特例.

Using the method originated by author in this paper,which is a method with neither increasing additional evaluationof function f(x,y)nor requiring bounded condition for f(x,y)and((?)^(i+j)f)/((?)x^i(?)y^j),2-stage second-order up to 4-stage fourth-orderRunge-Kutta formulas with minimizing the local truncation error are established.These formulas can be applied to non-linear systems of first-order ordinary differential equation,and are the improvement and extension of the results that aredue to Lotkin(1951),Ralston(1962),Merson(1957),Scraton(1964)mad England(1969).Moreover,when a sysytem isdegenerated into a single differential equation,Lotkin's(1951)and Ralston's(1964)some important results are the specialcases of this paper.