10.3. von Neumann stability analysis for the diffusion equation

The one-dimensional diffusion equation is

(10.5)\[\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2},\]

where \(D\) is the diffusivity.

The FTCS scheme applied to (10.5) is:

(10.6)\[u_m^{n+1} = u_m^{n} + D\frac{\Delta t}{\Delta x^2}(u_{m+1}^n-2u_{m}^n+u_{m-1}^n)\]

Substituting a solution like (4.6) in (10.6), we have

\[\begin{split}B^{n+1}e^{ikm\Delta x}=B^ne^{ikm\Delta x}+ D\frac{\Delta t}{\Delta x^2}B^{n} e^{ik(m+1)\Delta x}\\ -2B^{n} e^{ikm\Delta x}+B^{n} e^{ik(m-1)\Delta x}\end{split}\]

which, after some manipulation, allows to obtain the following expression for the amplification factor:

\[\frac{B^{n+1}}{B^n} = 1-2\tau+2\tau\cos k\Delta x = 1-4\tau \sin^2 \frac{k \Delta x}{2}, \quad \tau=D\frac{\Delta t}{\Delta x^2}.\]

The Von Neumann stability condition is:

(10.7)\[\left|\frac{B^{n+1}}{B^n}\right|\leq 1 \Leftrightarrow \left|1-4\tau \sin^2 \frac{k \Delta x}{2}\right| \leq 1, \quad \text{for all } k.\]

For \(\tau > 0\), the worst case occurs when \(\sin^2 \frac{k \Delta x}{2}=1\), which leads to

\[ \tau \leq \frac{1}{2}. \]

Therefore, the FTCS scheme apllied to the linear diffusion equation (10.5) is conditionally stable, with the stability condition

(10.8)\[D\frac{\Delta t}{\Delta x^2} \leq \frac{1}{2}.\]