CourantFriedrichsLewyKriteriums

The Courant-Friedrichs-Lewy (CFL) condition is a crucial principle in the numerical solution of partial differential equations, particularly those involving wave propagation. It establishes a fundamental limit on the time step size that can be used in explicit numerical methods. Essentially, the CFL condition states that the domain of dependence of the numerical solution at a point must contain the domain of dependence of the true solution of the differential equation. If this condition is violated, numerical instabilities will arise, leading to erroneous and often unbounded results.

This condition arises because information travels at a finite speed in many physical phenomena described by partial differential equations, such as fluid dynamics, electromagnetism, and acoustics. Numerical methods, especially explicit ones, approximate these solutions on a discrete grid. The CFL condition ensures that the numerical scheme can "capture" the propagation of information across grid cells within a given time step. If the time step is too large, a wave or disturbance can travel across multiple grid cells in a single time step, and the numerical method, which only considers information from adjacent cells at the previous time step, will not be able to accurately represent this movement.

The precise form of the CFL condition depends on the specific numerical method and the differential equation being solved. For simple one-dimensional advection equations, it often takes the form of $\Delta t \le C \frac{\Delta x}{v}$, where $\Delta t$ is the time step, $\Delta x$ is the spatial grid spacing, $v$ is the wave speed, and $C$ is a constant that depends on the numerical scheme (e.g., $C=1$ for first-order upwind). Violating the CFL condition can manifest as spurious oscillations or a complete breakdown of the simulation. Therefore, it is a critical parameter that must be carefully considered when designing and implementing numerical simulations.