# 交替方向隐式法

## 方法

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

${ u_{ij}^{n+1}-u_{ij}^n \over \Delta t } = {1 \over 2} \left( \delta_x^2 + \delta_y^2 \right) \left( u_{ij}^{n+1}+u_{ij}^n \right)$

ADI方法的思想在于将一个有限差分方程分割为两个，一个在x方向上隐式求导，另一个在y方向上隐式求导。

${u_{ij}^{n+1/2}-u_{ij}^n\over \Delta t/2} = \left(\delta_x^2 u_{ij}^{n+1/2}+\delta_y^2 u_{ij}^{n}\right)$
${u_{ij}^{n+1}-u_{ij}^{n+1/2}\over \Delta t/2} = \left(\delta_x^2 u_{ij}^{n+1/2}+\delta_y^2 u_{ij}^{n+1}\right).$

## 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, doi:10.1007/BF01386295, ISSN 0029-599X.
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, doi:10.1080/10407799108944957, ISSN 1040-7790.