Courant–Friedrichs–Lewy condition

In mathematics, the Courant–Friedrichs–Lewy condition (CFL condition) is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically by the method of finite differences.[1] It arises when explicit time-marching schemes 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 wildly incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.[2]

Contents

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 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 (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, the CFL condition is commonly prescribed for those terms of the finite-difference approximation of general partial differential equations which model the advection phenomenon.[5]

The one–dimensional case

For one-dimensional case, the CFL has the following form:

\frac {u \cdot \Delta\,t} {\Delta\,x} \leq C

where

  • u is the velocity (whose dimension is Length/Time)
  • Δt is the time step (whose dimension is Time)
  • Δx 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

\nu = \frac {u \cdot \Delta\,t} {\Delta\,x}

is called the Courant number.

The two and general n–dimensional case

In the two–dimensional case, the CFL condition becomes

\frac {u_ x \cdot \Delta\,t} {\Delta\,x} + \frac {u_ y \cdot \Delta\,t} {\Delta\,y} \leq C.

with obvious meaning of the symbols involved. In analogy with the two–dimensional case, the general CFL condition for the n–dimensional case is the following one

 {\Delta\, t}\cdot\sum_{i=1}^n\frac{u_{x_i}} {\Delta\, x_i} \leq C.

Note that the interval length it is not required to be the same for each spatial variable Δxi, 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

As already remarked, the CFL condition is 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 Δt: for example, in the finite-difference approximation of certain fourth-order nonlinear partial differential equations, it can have the following form

\frac{\Delta\, t}{(\Delta\, x)^4} < C_u,

meaning that a decrease in the length interval Δx requires a fourth order decrease in the time step Δt 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.

See also

Notes

  1. ^ In general, it is not a sufficient condition: also, it can be also a demanding condition for some problems. See the "Implications of the CFL condition" section of this article for a brief survey of this issues.
  2. ^ See reference Courant, Friedrichs & Lewy 1928. There exists also an English translation of the 1928 German original: see references Courant, Friedrichs & Lewy 1956 and Courant, Friedrichs & Lewy 1967.
  3. ^ This situation commonly occurs when a hyperbolic partial differential operator has been approximated by a finite difference equation, which is then solved by numerical linear algebra methods.
  4. ^ This quantity is not necessarily the same for for each spatial variable, as it is shown in the "The two and general n–dimensional case" section of this entry : it can be chosen in order to somewhat relax the condition.
  5. ^ Precisely, this is the hyperbolic part of the PDE under analysis.
  6. ^ Precisely, it does not depend on the time step Δt and on the length interval Δx.
  7. ^ See section 3 in the article by Roberts (2003)

References

External links


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