三维Helmholtz方程的高阶隐式紧致差分方法  

High-order Implicit Compact Difference Scheme for Solving the Three-dimensional Helmholtz Equation

作　　者：葛永斌[1] 刘国涛[2] 

机构地区：[1]宁夏大学应用数学与力学研究所,银川750021 [2]中国航天科工集团第六研究院46所,呼和浩特010010

出　　处：《工程数学学报》2010年第5期853-858,共6页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10502026);宁夏自然科学基金(NZ0937)~~

摘　　要：本文基于二阶导数的四阶Pade型紧致差分逼近式,并结合原方程本身,得到了三维Helmholtz方程的一种四阶精度的隐式紧致差分格式,该格式在每个空间方向上只涉及到三个点处的未知量及其二阶导数值。边界处对于二阶导数的离散格式利用四阶显式偏心格式。然后,利用Richardson外推法、算子插值法及二阶导数在边界点处的六阶显式偏心格式,将本文构造的格式精度提高到六阶。最后,通过数值实验验证了本文方法的精确性和可靠性。Based on the Pade scheme of second-order partial derivatives and combined with the original differential equation,a fourth-order implicit compact difference scheme is proposed for solving the three-dimensional Helmholtz equation.Only three points and their second-order derivative values are needed on each spatial direction.The fourth-order explicit difference schemes are used to construct the same order discretization of boundary conditions.Then,the accuracy of the fourth-order implicit compact difference scheme is upgraded to sixth-order by using the Richardson extrapolation technique and operator interpolation scheme.The sixth-order explicit difference schemes of second-order partial derivatives on the boundaries are also used.Finally,numerical experiments are given to prove the efficiency and reliability of the present method.

关 键 词：HELMHOLTZ方程 高精度 隐式 紧致差分格式 RICHARDSON外推法 

分 类 号：O241.82[理学—计算数学]