The CFL Condition
The stability limit that ties the time step of an explicit scheme to the grid spacing and the fastest wave speed.
Information cannot outrun the grid
The Courant-Friedrichs-Lewy (CFL) condition is a necessary stability requirement for explicit schemes applied to hyperbolic (wave-like) equations. Intuitively, in one time step physical information travels a distance equal to the wave speed times the step. If that distance exceeds the grid spacing, the numerical scheme cannot access the grid points that the true solution depends on, and it becomes unstable. The time step must therefore be small enough that information does not skip over cells.
The Courant number
The CFL condition is stated through the Courant number, defined as the wave speed times the time step divided by the grid spacing. For an explicit scheme it must stay below a scheme-dependent limit, often one. A Courant number of one means information travels exactly one cell per step; larger values break causality in the discrete scheme and amplify errors without bound.
Different equations, different scaling
- Advection and wave equations: the step scales linearly with grid spacing (dt proportional to dx)
- Diffusion equations, explicit: the step scales with the square of grid spacing (dt proportional to dx-squared), a far harsher limit
- Multiple wave speeds: the fastest wave in the system sets the limit
Living with or escaping the limit
In plasma simulation, fast waves (Alfven and magnetosonic) can impose punishing CFL limits on explicit MHD codes, forcing tiny steps. Two escapes exist: implicit schemes, which are unconditionally stable and free the step from the CFL limit at the cost of solving a system each step, and IMEX schemes, which treat only the fast, stiff waves implicitly. The CFL condition is thus the central reason implicit and semi-implicit methods dominate stiff wave-dominated simulation, and it must always be checked when a method-of-lines discretization uses an explicit integrator.