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Courant–Friedrichs–Lewy condition
In mathematics, the Courant–Friedrichs–Lewy (CFL) condition is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically by the method of finite differences.〔In general, it is not a sufficient condition; also, it can be a demanding condition for some problems. See the "Implications of the CFL condition" section of this article for a brief survey of these issues.〕 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 will produce incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.〔See reference . There exists also an English translation of the 1928 German original: see references and .〕
==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,〔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.〕 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) in order to assure that the scheme can access the information required to form the solution.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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