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The CFL (Courant) Condition

Equation

For explicit wave-propagation schemes on a grid (FDTD):

\[\boxed{\,\Delta t \leq \frac{\Delta x}{c\sqrt{d}}\,}\]

\(d\) = number of spatial dimensions (1, 2, 3); \(c\) = the fastest signal speed in the system.

Physical meaning

Causality on the grid. In one time step, the numerical scheme moves information at most one cell (\(\Delta x\)); physics moves it \(c\,\Delta t\). If physics outruns the stencil (\(c\Delta t > \Delta x/\sqrt d\)), the numerical domain of dependence cannot contain the true one — the scheme is being asked to predict without the needed information, and errors amplify explosively. Named for Courant, Friedrichs & Lewy (1928) — a stability law older than the computer.

Variables

\(\Delta t\) — time step · \(\Delta x\) — grid spacing · \(c\) — wave/signal speed · \(\sqrt d\) — the diagonal factor (waves can travel diagonally across cells).

Consequences

  • Cost scaling: refining space by 2 forces refining time by 2 → cost \(\times 2^{d+1}\) in \(d\) dimensions.
  • Fast, irrelevant waves still bind you. If light waves live in your model but you care about slow plasma waves, explicit stepping still requires resolving \(c\) — a key motivation for implicit and reduced models.
  • Practical safety margin: the course uses \(\Delta t = \Delta x/2c\) (1-D) and \(\Delta t = \frac{\sqrt2}{2}\Delta x/c\) (2-D).

Assumptions / scope

Applies to explicit hyperbolic (wave-type) schemes. Diffusion equations have their own, harsher constraint (\(\Delta t \lesssim \Delta x^2/2\kappa\)); implicit methods trade the constraint for a linear solve; the ODE analogue is the stability limit of forward Euler.

Quiz

Q1 (computational). 1-D FDTD with \(\Delta x = 1\) mm: what is the largest stable \(\Delta t\) for light?

Answer

\(\Delta t \leq \Delta x/c = 10^{-3}/3\times10^8 \approx 3.3\) ps.

Q2 (conceptual). Violating CFL doesn't add a little error — the run explodes within tens of steps. Why so violent?

Answer

Instability is multiplicative: the offending Fourier modes are amplified by a factor \(|g| > 1\) every step, so errors grow like \(|g|^n\) — exponential blow-up, usually fastest at the grid's shortest wavelength (checkerboard noise).

Q3 (MCQ). In 3-D, the CFL limit relative to 1-D (same \(\Delta x\), \(c\)) is:

  • (a) the same (b) \(\sqrt3\) smaller (c) 3× smaller (d) 3× larger
Answer

(b). \(\Delta t \leq \Delta x/(c\sqrt3)\) — diagonal propagation across the cell sets the pace.