常微方程组初值问题的连续有限元及守恒性  

CONTINUOUS FINITE ELEMENT AND ITS CONSERVATION FOR INITIAL VALUE PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

作　　者：刘罗华[1] 汤琼[1] 杨禄源[1] 

机构地区：[1]株洲工学院信息与计算科学系,株洲412008

出　　处：《高等学校计算数学学报》2005年第2期178-184,共7页Numerical Mathematics A Journal of Chinese Universities

摘　　要：1引言 常微分方程数值解出现在科学与工程计算的许多领域,如生物繁殖、自动控制、卫星轨道等.目前已有许多数值解法:Euler法、梯形法、Runge Kutta法、多步法等.将偏微分方程中有限元思想用于解常微分方程,此思想在国外有Delfour,Hager,Aziz等人,见[1][2].国内有陈传淼、潘青等人进行过研究,见[3],[4].本文利用对偶论证的方法证明了常微分方程的连续有限元的超收敛性及在节点上的超收敛性,并讨论了相应的单步κ级隐格式的稳定性,得出用任意r次连续有限元求解Hamilton方程组时其能量是守恒的.In this paper the continous finite element to solve initinal value problem for system of linear differential equations is used, and the absolute stability of the corresponding single step k-order hidden shceme is discussed. In the paper by simplified means, the superconvergence of finite element and one of it on the nodes are proved. Using the continuous finite element to solve linear Hamilton systems: pt = Hq,qt =-Hp, the conservation of energy H(p,q) = -1/2ap2 + bpq +1/2cq2 can be obtained. The computation shows that even if division is regular and the error of finite element ph,qh is big, H(ph,qh) is almost equal H(p,q) in the range of computation accuracy.

关 键 词：常微方程组 初值问题 连续有限元法 守恒性 

分 类 号：O241.81[理学—计算数学] O175.8[理学—数学]