1 什么叫CFL数？
CFL数是收敛条件，具体是差分方程的依赖域必须包含相应微分方程的依赖域，最简单可以理解为时间推进求解的速度必须大于物理扰动传播的速度，只有这样才能将物理上所有的扰动俘获到。Time stepping technique是指时间推进技术，一般有统一时间步长和当地时间步长，而选择当地时间步长也就是当地CFL条件允许的最大时间步长，采用这种方法能够加速收敛，节省计算时间。R
CFL条件的来历
在有限差分和有限体积方法中的稳定性和收敛性分析中有一个很重要的概念------CFL条件。CFL条件是以Courant,Friedrichs,Lewy三个人的名字命名的，他们最早在1928年一篇关于偏微分方程的有限差分方法的文章中首次踢出这个概念的时候，并不是用来分析差分格式的稳定性，而是仅仅以有限差分方法作为分析工具来证明某些偏微分方程的解的存在性的。其基本思想是先构造PDE的差分方程得到一个逼近解的序列，只要知道在给定的网格系统下这个逼近序列收敛，那么久很容易证明这个收敛解就是愿微分方程的解。Courant,Friedrichs,Lewy发现，要使这个逼近序列收敛，必须满足一个条件，那就是著名的
CFL条件，记述如下：
CFL condition:An numerical method can be convergent only if its numerical
domain of dependence contains the true domain of dependence of the PDE,
at least in the limit as dt and dx go to zero.
随着计算机的迅猛发展，有限差分方法和有限体积方法越来越多的应用于流体力学的数值模拟中，CFL条件作为一个格式稳定性和收敛性的判据，也随之显得非常重要了。但值得注意的是，CFL条件仅仅是稳定性(收敛性)的必要条件，而不是充分条件，举例来说，数值流通量构造方法中的算术平均构造，它在dt足够小的情况下是可以满足CFL条件，但对于双曲问题而言这种构造方法是不稳定，不可用的。
在双曲问题的现格式方法中，一般取CFL数小于1且在1附近的值，这样沿特征线的传播不至于偏离得太远或者太近，进而可以保证数值解得准确性。
在抛物型问题中对CFL条件的要求要来得更加严格，因为在下一个时间层上的任意一点上
的影响域是所有时间层上所有离散点。怎样在差分格式中体现抛物型问题的这样一个特点呢？一般对于显式格式，可以取时间步长dt=O(dx~2)；更好的方法是采用隐式格式。
Courant–Friedrichs–Lewy condition
From Wikipedia, the free encyclopedia
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 aris in the numerical analysis of explicit time-marching schemes, when the are ud for the numerical solution. As a conquence, the time step must be less than a certain time in many explicit time-marchingcomputer simulations, otherwi 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 paration is reduced, the upper limit for the time step also decreas. In esnce, 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 preci 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 pod.
∙Spatial dimension of the problem: it is the number of spatial dimensions i.e. the number of spatial coordinates of the physical space where the problem is pod. Typical values are , and .
∙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 suppod to be discrete valued independent variables, who 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 tho terms of the finite-difference approximation of general partial differential equations which model the advection phenomenon.[5]
The one-dimensional ca
For one-dimensional ca, the CFL has the following form:
where the dimensionless number is called the Courant number,
∙ is the velocity (who dimension is length/time)
∙ is the time step (who dimension is time)
∙ is the length interval (who dimension is length).
The value of changes with the method ud to solve the discretid equation. If an explicit (time-marching) solver is ud then typically . Implicit (matrix) solvers are usually less nsitive to numerical instability and so larger values of may be tolerated.
The two and general n-dimensional ca
In the two-dimensional ca, the CFL condition becomes
