CFL条件.docx

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(dx2)；更好的方法是采用隐式格式。CourantFriedrichsLewy conditionFrom Wikipedia, the free encyclopediaInmathematics, theCourantFriedrichsLewy condition(CFL condition) is a necessary condition for convergence while solving certainpartial differential equations(usuallyhyperbolic PDEs) numerically by themethod of finite differences.1It arises in thenumerical analysisofexplicittime-marching schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in manyexplicittime-marchingcomputer simulations, otherwise the simulation will produce incorrect results. The condition is named afterRichard Courant,Kurt Friedrichs, andHans Lewywho described it in their 1928 paper.2Heuristic descriptionThe information behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute itsamplitudeat discrete time steps of equal length,3then 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. 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 conditionIn 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 thecoordinatesof thephysical spacein which the problem is posed. Spatial dimension of the problem: it is the numberofspatial dimensionsi.e. the number of spatialcoordinatesof thephysical spacewhere the problem is posed. Typical values are,and. Time: it is thecoordinate, acting as aparameter, which describes the evolution of the system, distinct from the spatial coordinates.The spatial coordinates and the time are supposed to bediscrete valued independent variables, whose minimal steps are called respectively theinterval length4and thetime 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 thefinite-difference approximationof generalpartial differential equationswhich model theadvectionphenomenon.5The one-dimensional caseFor one-dimensional case, the CFL has the following form:where thedimensionless numberis called theCourant number, is the velocity (whosedimensionis length/time) is the time step (whosedimensionis time) is the length interval (whosedimensionis length).The value ofchanges with the method used to solve the discretised equation. If an explicit (time-marching) solver is used then typically. Implicit (matrix) solvers are usually less sensitive to numerical instability and so larger values ofmay be tolerated.The two and generaln-dimensional caseIn thetwo-dimensionalcase, the CFL condition becomeswith obvious meaning of the symbols involved. By analogy with the two-dimensional case, the general CFL condition for the-dimensional case is the following one:The interval length is not required to be the same for each spatial variable. 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 conditionThe CFL condition is only a necessary oneThe CFL condition is a necessary condition, but may not be sufficient for the convergence of thefinite-difference approximationof a givennumerical 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 requirementThe CFL condition can be a very limiting constraint on the time step: for example, in thefinite-difference approximationof certain fourth-order nonlinear partial differential equations, it can have the f