Von Neumann Stability Analysis

The von Neumann stability analysis method is simple to apply but it cannot handle boundary conditions.

It consists of replacing the spatial variation by a single Fourier component. The method is sufficient for linear equations with constant coefficients.

We shall illustrate the Von Neumann stability method with the FTCS scheme.

Stability of the FTCS scheme

The linear advection equation

The FTCS scheme for the linear advection equation is given by:

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

To apply the Von Neumann stability analysis method we assume a solution of the form:

(39)\[ u_m^n=B^n e^{ikm\Delta x}. \]

Substituing in (38), we get

\[\begin{split} B^{n+1} e^{ikm\Delta x} &= B^{n} e^{ikm\Delta x} - c\frac{\Delta t}{2\Delta x}(B^{n} e^{ik(m+1)\Delta x}-B^{n} e^{ik(m-1)\Delta x}) \\ B^{n+1} e^{ikm\Delta x} &= B^{n} e^{ikm\Delta x}\left[1 - c\frac{\Delta t}{2\Delta x}(e^{ik\Delta x}-e^{-ik\Delta x})\right]. \end{split}\]

Eliminating the common factor \(e^{ikm\Delta x}\) and defining the amplification factor as \(|B^{n+1}/B^n|\), we can write:

\[ \frac{B^{n+1}}{B^n}=1-c\frac{\Delta t}{\Delta x}i\sin k\Delta x \]

For the scheme to be stable, we require that the amplification factor be \(\leq 1\):

\[ \left|\frac{B^{n+1}}{B^n}\right|=\left|1-c\frac{\Delta t}{\Delta x}i\sin k\Delta x\right|\leq 1, \]

which is impossible to fulfill, because

\[ \left|\frac{B^{n+1}}{B^n}\right|=\sqrt{1+\left(c\frac{\Delta t}{\Delta x}\right)\sin^2 k\Delta x} \ge 1, \quad \text{for all } k. \]

Therefore, the FTCS scheme applied to the linear advection equation is always unstable, i.e., it is unconditionally unstable.

The diffusion equation

The one-dimensional diffusion equation is

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

where \(D\) is the diffusivity.

The FTCS scheme applied to (40) is:

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

Substituting a solution like (39) in (41), 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:

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

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 (40) is conditionally stable, with the stability condition

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