In mathematics, the Courant–Friedrichs–Lewy condition (CFL condition) isa necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically by the method of finite differences.[1]
雖然在Citcom解地幔對流中，每一個時間步是有限元法，但是在時間演化上，仍然是有限差分。我現在，還沒見到過有限元用在時間演化（時間導數）上的。
It arises when explicit time-marching schemes（顯式時間推進計划，顯式格式條件穩定和條件收斂，而隱式格式往往是無條件穩定和無條件收斂的，但是不容易求解數值解。）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 will produce incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.[2]
Heuristic description
The information 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 length,[3] then this length 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. （當空間網格的長度減小時，時間步的長度也應該減小，恩，怪不得當我Citcom中，把空間步長減小時，時間演化就會減慢。原來是在這里啊）In essence, the numerical domain of dependence of any point in space and time (which data values in the initial conditions affect the numerical computed value at that point) must include the analytical domain of dependence (where in the initial conditions has an effect on the exact value of the solution at that point) in order to assure that the scheme can access the information required to form the solution.
The CFL condition
In order to make a reasonably formally precise statement of the condition, it is necessary to define the following quantities
Spatial coordinate: it is one of the coordinates of the physical space in which the problem is posed.
Spatial dimension of the problem: it is the number 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, n=2 and n=3.
Time: it is 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 supposed to be discrete valued independent variables, whose minimal steps are called respectively the interval length[4] and the time step: the CFL condition relates the length of the time step to a function interval lengths of each spatial variable.
Operatively, theCFLcondition is commonly prescribed for those terms of the finite-difference approximation of general partial differential equations which model theadvection phenomenon.[5]
The one-dimensional case
For one-dimensional case, the CFL has the following form:
where
u is the velocity (whose dimension is Length/Time)
\Delta t is the time step (whose dimension is Time)
\Delta x is the length interval (whose dimension is Length),
C is a dimensionless constant which depends only on the particular equation to be solved.[6]
The dimensionless number
is called the Courant number.
例如，對於波動方程而言，niu我一般取0.1，當然，其實niu=0.5就行了，但是，演化不了多少步就會發散。
The two and general n-dimensional case
In the two-dimensional case, the CFL condition becomes
with obvious meaning of the symbols involved. By analogy with the two-dimensional case, the general CFL condition for the n-dimensional case is the following one:
Note that the interval length it is not required to be the same for each spatial variable \Delta x_i, i =1, ..., n. This "degree of freedom" can be used in order to somewhat optimize the value of the time step for a particular problem, by varying the values of the different interval in order to keep it not too small.
Implications of the CFL condition
The CFL condition is only a necessary one
只是必要條件，並非充分條件
As already remarked, the CFL conditionis a necessary condition, but may not be sufficient for the convergence of the Finite-difference approximation of a given numerical problem. Thus, in order to establish the convergence of the finite-difference approximation, it is necessary to use other methods, which in turn could imply further limitations on the length of the time step and/or the lengths of the spatial intervals.
The CFL condition can be a very strong requirement
The CFL condition can be a very limiting constraint on the time step \Delta t: for example, in the finite-difference approximation of certain fourth-order nonlinear partial differential equations, it can have the following form
meaning that a decrease in the length interval \Delta x requires a fourth order decrease in the time step \Delta t for the condition to be fulfilled. Therefore, when solving particularly stiff problems, efforts are often made to avoid the CFL condition, for example by using implicit methods. However, in a recent work,[7] a modern dynamical systems approaches to modeling, based upon center manifold theory, is demonstrated to provide theoretical support for the construction of non-traditional discretisations that automatically overcome the CFL restriction: see the article by Roberts (2003) for further information.