# 交替方向隐式法

## 方法

${\displaystyle {\partial u \over \partial t}=\left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}\right)=(u_{xx}+u_{yy})\quad }$

${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n} \over \Delta t}={1 \over 2}\left(\delta _{x}^{2}+\delta _{y}^{2}\right)\left(u_{ij}^{n+1}+u_{ij}^{n}\right)}$

${\displaystyle {u_{ij}^{n+1/2}-u_{ij}^{n} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n}\right)}$
${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n+1/2} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n+1}\right).}$

## 参考文献

1. ^ Peaceman, D. W.; Rachford Jr., H. H., The numerical solution of parabolic and elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 1955, 3: 28–41, MR0071874.
2. ^ Douglas, Jr., J., On the numerical integration of uxx+ uyy= utt by implicit methods, Journal of the Society of Industrial and Applied Mathematics, 1955, 3: 42–65, MR0071875.
3. ^ Douglas Jr., Jim, Alternating direction methods for three space variables, Numerische Mathematik, 1962, 4: 41–63, ISSN 0029-599X, doi:10.1007/BF01386295.
4. ^ Chang, M. J.; Chow, L. C.; Chang, W. S., Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems, Numerical Heat Transfer, Part B: Fundamentals, 1991, 19 (1): 69–84, ISSN 1040-7790, doi:10.1080/10407799108944957.