Courant–Friedrichs–Lewy condition

In mathematics, the convergence condition by Courant–Friedrichs–Lewy (CFL) is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain upper bound, given a fixed spatial increment, in many explicit time-marching computer simulations; otherwise, the simulation produces incorrect or unstable results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.

Heuristic description

The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for the time step also decreases. In essence, the numerical domain of dependence of any point in space and time (as determined by initial conditions and the parameters of the approximation scheme) must include the analytical domain of dependence (wherein the initial conditions have an effect on the exact value of the solution at that point) to assure that the scheme can access the information required to form the solution.

Statement

To make a reasonably formally precise statement of the condition, it is necessary to define the following quantities:

Spatial coordinate: one of the coordinates of the physical space in which the problem is posed

Spatial dimension of the problem: the number

n

{\displaystyle n}

of spatial dimensions, i.e., the number of spatial coordinates of the physical space where the problem is posed. Typical values are

n

=

1

{\displaystyle n=1}

,

n

=

2

{\displaystyle n=2}

and

n

=

3

{\displaystyle n=3}

.

Time: the coordinate, acting as a parameter, which describes the evolution of the system, distinct from the spatial coordinates

The spatial coordinates and the time are discrete-valued independent variables, which are placed at regular distances called the interval length and the time step, respectively.