Courant–Friedrichs–Lewy Condition

The CFL condition is a fundamental concept governing the stability of explicit time-marching numerical methods. It arises from the principle that the numerical domain of dependence of a grid point must contain the physical domain of dependence. In simpler terms, for a calculation at a grid point (i) at the next time step (n+1), the numerical scheme uses information from neighboring points at the current time step (n). The CFL condition ensures that any physical phenomenon (like a pressure wave) that could have reached point (i) in the time interval \(\Delta t\) must have originated from within that set of neighboring points.

In the formula \(C = \frac{u \Delta t}{\Delta x} \le C_{max}\), \(C\) is the dimensionless Courant number, \(u\) is the maximum wave propagation speed in the system (e.g., fluid velocity plus the speed of sound for compressible flow), \(\Delta t\) is the time step, and \(\Delta x\) is the grid spacing. The value of \(C_{max}\) depends on the specific numerical scheme but is often on the order of 1. If the condition is violated (\(C > C_{max}\)), the numerical solution becomes unstable, with errors growing exponentially, leading to a non-physical, divergent result. This imposes a severe restriction on the time step size, especially in meshes with very fine cells (\(\Delta x\) is small), making explicit methods computationally expensive for certain problems. Implicit methods, while more complex per time step, are often unconditionally stable and not subject to the CFL constraint, allowing for much larger time steps.