The one-dimensional case

For the one-dimensional case, the continuous-time model equation (that is usually solved for ${\displaystyle w}$) is:

${\displaystyle {\frac {\partial w}{\partial t}}=u{\frac {\partial w}{\partial x}}.}$

The CFL condition then has the following form:

${\displaystyle C={\frac {u\,\Delta t}{\Delta x}}\leq C_{\max }}$

where the dimensionless number ${\displaystyle C}$ is called the Courant number,

• ${\displaystyle u}$ is the magnitude of the velocity (whose dimension is length/time)
• ${\displaystyle \Delta t}$ is the time step (whose dimension is time)
• ${\displaystyle \Delta x}$ is the length interval (whose dimension is length).

The value of ${\displaystyle C_{\max }}$ changes with the method used to solve the discretised equation, especially depending on whether the method is explicit or implicit. If an explicit (time-marching) solver is used then typically ${\displaystyle C_{\max }=1}$. Implicit (matrix) solvers are usually less sensitive to numerical instability and so larger values of ${\displaystyle C_{\max }}$ may be tolerated.

The two and general n-dimensional case

In the two-dimensional case, the CFL condition becomes

${\displaystyle C={\frac {u_{x}\,\Delta t}{\Delta x}}+{\frac {u_{y}\,\Delta t}{\Delta y}}\leq C_{\max }}$

with the obvious meanings of the symbols involved. By analogy with the two-dimensional case, the general CFL condition for the ${\displaystyle n}$-dimensional case is the following one:

${\displaystyle C=\Delta t\left(\sum _{i=1}^{n}{\frac {u_{i}}{\Delta x_{i}}}\right)\leq C_{\max }.}$

The interval length is not required to be the same for each spatial variable ${\displaystyle \Delta x_{i},i=1,\ldots ,n}$. 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.