利用对称网格点泰勒展开式中各阶导数项明显的对称性,得到了数值求解三维泊松方程的四阶和六阶精度的肾致差分格式,其推导过程简便直接.为了克服传统迭代法在求解高维问题时计算量大、收敛速度慢的缺陷,采用了多重网格加速技术,设计了相应的多重网格算法,求解了三维泊松方程的Dirichlet边值问题.数值实验结果表明,本文所提出的高精度肾致格式达到了期望的精度并且多重网格方法的加速效果是非常显著的.
Based on the Taylor expansion,a fourth order and a sixth-order compact difference scheme for the three dimensional (3 D) Poisson equation are presented. A muhigrid method and multigrid accelerating technique are employed to overcome the difficulties (larger cost and lower speed of convergence) when traditional relaxation methods are used to treat high dimensional problems. By using the method and the technique,the boundary value problems of the 3-D Poisson equation are solved. Numerical experiments results shown that the new high order compact difference schemes fit our expectation and the multigrid algorithm is very efficient.