Stability and the CFL Condition

A Hard Speed Limit

Marching the wave equation forward with an explicit stencil is fast, but it comes with a non-negotiable rule. In a single time step, information may only travel a fraction of a grid cell. Break that rule and the scheme does not merely lose accuracy; it becomes unstable and the wavefield explodes. The controlling quantity is the Courant number

$C = \dfrac{c\,\Delta t}{h},$

the distance a wave travels in one step measured in grid cells. The CFL condition (Courant, Friedrichs, Lewy) requires $C$ to stay below a limit set by the scheme: $C \le 1$ in one dimension, and $C \le 1/\sqrt{2} \approx 0.71$ for the two-dimensional engine.

Below the Line and Above It

Slide the Courant number and the two panels tell the whole story. Below the limit the space-time image shows a clean pulse tracing its path and the amplitude curve stays flat. Cross the limit and a checkerboard instability erupts, doubling every few steps until it saturates everything; the amplitude curve turns into a rising exponential. There is no gentle degradation, just a cliff.

The practical consequence shapes every large simulation. Because $\Delta t \le h / c$ (up to the constant), faster rock forces a smaller time step, and so does a finer grid. A high-velocity salt body or a high-resolution model can dominate the cost of a run, not through more grid points alone but through the tiny time step the fastest cell demands. This is why the engine ships a cflDt helper that returns a safe step below the limit: it is cheaper to compute the right time step once than to watch a simulation blow up. With stability secured, the remaining error is the subtle one this part builds toward, numerical dispersion.