# 冯诺依曼稳定性分析

## 方法描述

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

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

${\displaystyle \quad (2)\qquad \epsilon _{j}^{n+1}=\epsilon _{j}^{n}+r\left(\epsilon _{j+1}^{n}-2\epsilon _{j}^{n}+\epsilon _{j-1}^{n}\right)}$

${\displaystyle \quad (3)\qquad \epsilon (x)=\sum _{m=1}^{M}A_{m}e^{ik_{m}x}}$

${\displaystyle \quad (4)\qquad \epsilon (x,t)=\sum _{m=1}^{M}e^{at}e^{ik_{m}x}}$

${\displaystyle \quad (5)\qquad \epsilon _{m}(x,t)=e^{at}e^{ik_{m}x}.}$

• {\displaystyle {\begin{aligned}\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{aligned}}}

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

${\displaystyle \qquad \cos(k_{m}\Delta x)={\frac {e^{ik_{m}\Delta x}+e^{-ik_{m}\Delta x}}{2}}}$ ${\displaystyle \sin ^{2}{\frac {k_{m}\Delta x}{2}}={\frac {1-\cos(k_{m}\Delta x)}{2}}}$

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

${\displaystyle G\equiv {\frac {\epsilon _{j}^{n+1}}{\epsilon _{j}^{n}}},}$

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

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

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

(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 ${\displaystyle \Delta x}$  ， 所允许的 ${\displaystyle \Delta t}$  取值需要足够小以满足 (10) ，才能保证计算的数值稳定。

## 参考资料

1. ^ Analysis of Numerical Methods by E. Isaacson, H. B. Keller. [2011-05-20]. （原始内容存档于2011-05-21）.
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.