# 冯诺依曼稳定性分析

## 方法描述

$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

$\quad (1) \qquad u_j^{n + 1} = u_j^{n} + r \left(u_{j + 1}^n - 2 u_j^n + u_{j - 1}^n \right)$

$\quad (2) \qquad \epsilon_j^{n + 1} = \epsilon_j^n + r \left(\epsilon_{j + 1}^n - 2 \epsilon_j^n + \epsilon_{j - 1}^n \right)$

$\quad (3) \qquad \epsilon(x) = \sum_{m=1}^{M} A_m e^{ik_m x}$

$\quad (4) \qquad \epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x}$

$\quad (5) \qquad \epsilon_m(x,t) = e^{at} e^{ik_m x}.$

• \begin{align} \epsilon_j^n & = e^{at} e^{ik_m x} \\ \epsilon_j^{n+1} & = e^{a(t+\Delta t)} e^{ik_m x} \\ \epsilon_{j+1}^n & = e^{at} e^{ik_m (x+\Delta x)} \\ \epsilon_{j-1}^n & = e^{at} e^{ik_m (x-\Delta x)}, \end{align}

$\quad (6) \qquad e^{a\Delta t} = 1 + \frac{\alpha \Delta t}{\Delta x^2} \left(e^{ik_m \Delta x} + e^{-ik_m \Delta x} - 2\right).$

$\qquad \cos(k_m \Delta x) = \frac{e^{ik_m \Delta x} + e^{-ik_m \Delta x}}{2}$$\sin^2\frac{k_m \Delta x}{2} = \frac{1 - \cos(k_m \Delta x)}{2}$

$\quad (7) \qquad e^{a\Delta t} = 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2).$

$G \equiv \frac{\epsilon_j^{n+1}}{\epsilon_j^n},$

$\quad (8) \qquad G = \frac{e^{a(t+\Delta t)} e^{ik_m x}}{e^{at} e^{ik_m x}} = e^{a\Delta t},$

$\quad (9) \qquad \left\vert 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2) \right\vert \leq 1$

$\quad (10) \qquad \frac{\alpha \Delta t}{\Delta x^2} \leq \frac{1}{2}.$

(10) 即为该算法的稳定性条件。 对于 FTCS 求解一维热传导方程，给定 $\Delta x$ ， 所允许的 $\Delta t$ 取值需要足够小以满足 (10) ，才能保证计算的数值稳定。

## 參考資料

1. ^ Analysis of Numerical Methods by E. Isaacson, H. B. Keller
2. ^ Crank, J.; Nicolson, P., A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of Heat Conduction Type, Proc. Camb. Phil. Soc., 1947, 43: 50–67, doi:10.1007/BF02127704
3. ^ Charney, J. G.; Fjørtoft, R.; von Neumann, J., Numerical Integration of the Barotropic Vorticity Equation, Tellus, 1950, 2: 237–254
4. ^ Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., 67–68, 1985
5. ^ Anderson, J. D., Jr.. Computational Fluid Dynamics: The Basics with Applications. McGraw Hill. 1994.