The CFL Condition
The Courant-Friedrichs-Lewy condition bounds the explicit time step by requiring the numerical domain of dependence to contain the physical one.
Information cannot outrun the grid
The Courant-Friedrichs-Lewy (CFL) condition is a necessary stability requirement for explicit schemes on hyperbolic problems. It states that in one time step, a physical signal must not travel farther than the numerical stencil can reach. If the true wave crosses more than one cell per step while the scheme only samples neighboring cells, the scheme lacks the information to represent the physics and becomes unstable.
Quantitatively, the Courant number C = a*dt/dx, where a is the wave speed, dx the cell size, and dt the step, must not exceed a scheme-dependent limit, often one for simple explicit upwind schemes.
Domains of dependence
The condition is stated precisely in terms of domains of dependence. The physical domain of dependence of a point is the set of earlier data that can influence it; the numerical domain of dependence is the set of grid values the scheme actually uses. Stability requires the numerical domain to contain the physical one, so that the scheme has access to all data the true solution depends on.
Consequences and multi-scale problems
The CFL bound couples time and space resolution: refining the grid forces proportionally smaller time steps. For systems with widely varying wave speeds, the fastest wave dictates the step for the whole domain, which can make explicit integration inefficient when the fast wave is physically unimportant. This is a primary motivation for implicit and IMEX methods, which relax or remove the constraint for the stiff waves.
- Courant number C = a*dt/dx must stay below a scheme limit
- Numerical domain of dependence must contain the physical one
- Grid refinement forces proportional time-step reduction
- Fastest wave sets the step, motivating implicit methods
Every explicit fusion simulation manages its time step against the CFL limit, whether set by Alfven and magnetosonic waves in MHD or by particle and light speeds in kinetic models.