Von Neumann Stability Analysis
Von Neumann analysis tests a scheme's stability by tracking how individual Fourier modes are amplified over a time step.
Fourier modes as probes
Von Neumann stability analysis determines whether a linear finite-difference scheme keeps errors bounded. It decomposes the numerical error into Fourier modes and asks how each mode's amplitude changes over one time step. Substituting a single mode of the form exp(i k x) into the scheme yields an amplification factor g(k): the ratio of the mode's amplitude after a step to its amplitude before.
The scheme is stable if the magnitude of g(k) does not exceed one for every wavenumber k the grid can represent (allowing a small growth proportional to the step for problems with genuine growth). Any mode with |g| greater than one grows without bound and destroys the solution.
What it reveals
The analysis directly produces stability limits. For the explicit heat equation it yields the condition that the diffusion number be at most one half, forcing dt to scale with dx squared. For advection it recovers the CFL bound. It also reveals numerical dispersion and dissipation: the phase and magnitude of g(k) show how different wavelengths propagate and decay compared with the true solution.
Scope and limitations
Von Neumann analysis assumes a linear, constant-coefficient problem on a periodic or infinite domain, so it strictly applies to that idealization. Boundaries, variable coefficients, and nonlinearity can change stability, but the analysis remains the essential first test and a reliable guide for the interior scheme. It also connects to the Lax equivalence theorem: consistency plus stability implies convergence.
- Amplification factor g(k) tracks per-mode growth
- Stability requires |g(k)| <= 1 for all resolved wavenumbers
- Recovers CFL and diffusion-number step limits
- Assumes linear, constant-coefficient, periodic problems
This analysis is the standard tool for setting time-step limits and diagnosing dispersion and dissipation errors when designing discretizations for transport and MHD codes.