CourantFriedrichsLewy

Courant-Friedrichs-Lewy (CFL) condition is a fundamental concept in the field of partial differential equations and numerical analysis. It is a necessary condition for the stability and accuracy of numerical methods used to solve hyperbolic partial differential equations (PDEs).

In 1928, Rudolf Courant, Kurt Friedrichs, and Hans Lewy published a paper introducing the concept of CFL condition, while working on solving the neutron transport equation. They realized that numerical methods used to solve hyperbolic PDEs must satisfy a particular condition to ensure convergence and accuracy. This condition is now known as the Courant-Friedrichs-Lewy condition.

The CFL condition is based on the relationship between the numerical time step and the spatial grid size. Specifically, it states that the time step must be smaller than or equal to a certain fraction of the spatial grid size, which is determined by the speed of propagation of the hyperbolic waves. Mathematically, this is expressed as dx/dt <= c, where dx is the spatial grid size, dt is the time step, and c is the speed of propagation.

The CFL condition is not only a necessary condition for stability but also a sufficient condition for accuracy. If the CFL condition is not satisfied, the numerical solution may exhibit oscillations, numerical instability, or other undesirable behavior. On the other hand, if the CFL condition is met, the numerical solution will be stable and accurate. The CFL condition has far-reaching implications for numerical analysis, particularly in the context of fluid dynamics, plasma physics, and other areas that rely heavily on numerical methods to solve hyperbolic PDEs.