解两点边值问题的精细循环约化方法  

A precise cyclic reduction method for solving linear two-point boundary value problems

作　　者：富明慧[1] 陈焯智[1] 

机构地区：[1]中山大学,广州510275

出　　处：《应用力学学报》2012年第5期573-578,630,共6页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(11172334);2010年广东省大学生创新实验项目(1055810002)

摘　　要：将精细积分技术与循环约化方法相结合,提出两点边值问题的一种高精度、高效率求解方法。将求解域均匀离散,利用相邻两点间的传递关系式建立区段代数方程,将各区段的代数方程集成代数方程组,并利用循环约化方法对其求解。由于离散过程中几乎没有引入离散误差,并且在循环约化过程中采用了大量、小量分离技术,因此本方法具有极高的精度;同时循环约化过程充分利用2N算法的特点,使计算效率高、存储量小。研究结果表明,相对于已有的求解两点边值问题的精细积分法,本文方法适用范围更广,效率更高。例如对两端固支、受均布横向荷载作用下梁的非齐次方程计算,本文方法的精度可达到小数点后十三位,已经非常精确。Combining the techniques of precise integration method(PIM) and the cyclic reduction method(CRM),a highly precise and effective method for solving the linear two-point boundary value problem(TPBVP) is introduced.The solving domain is uniformly discretized,while the interval algebraic equations are created by employing the transition relationship between two adjacent points.All equations can then be solved by the CRM.Because almost no discretization error is introduced and the technique of separating the major value and minimum value is applied,highly precise results can be obtained.Meanwhile,this method utilizes the merit of 2N algorithm,which increases the effectiveness and saves the computational resource.Compared to the conventional PIM methods for solving the TPBVP,the method implemented is more effective and can be more widely employed,as demonstrated by the numerical examples.For instance,in Example Three,13 digits after the decimal point is the same as the exact solution.

关 键 词：两点边值问题 精细积分法 循环约化方法 大量小量分离技术 非齐次方程 

分 类 号：O302[理学—力学]