Courant–Friedrichs–Lewy condition


In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations 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 time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect 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 must include the analytical domain of dependence 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:
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. Using these names, the CFL condition relates the length of the time step to a function of the interval lengths of each spatial coordinate and of the maximum speed that information can travel in the physical space.
Operatively, the CFL condition is commonly prescribed for those terms of the finite-difference approximation of general partial differential equations that model the advection phenomenon.

The one-dimensional case

For one-dimensional case, the CFL has the following form:
where the dimensionless number is called the Courant number,
The value of changes with the method used to solve the discretised equation, especially depending on whether the method is explicit or implicit. If an explicit solver is used then typically. Implicit solvers are usually less sensitive to numerical instability and so larger values of may be tolerated.

The two and general ''n''-dimensional case

In the two-dimensional case, the CFL condition becomes
with the obvious meanings of the symbols involved. By analogy with the two-dimensional case, the general CFL condition for the -dimensional case is the following one:
The interval length is not required to be the same for each spatial variable. This "degree of freedom" can be used to somewhat optimize the value of the time step for a particular problem, by varying the values of the different interval to keep it not too small.