4.5. 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.

4.5.1. Stability of the FTCS scheme

4.5.1.1. The linear advection equation

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

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

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

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

Substituing in (4.5), 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.