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| In [[mathematics]], the '''Courant–Friedrichs–Lewy condition''' (CFL condition) is a necessary condition for [[Numerical stability#Stability in numerical differential equations|stability]] while solving certain [[partial differential equation]]s (usually [[Hyperbolic partial differential equation|hyperbolic PDE]]s) numerically by the [[Finite difference method|method of finite differences]].<ref>In general, it is not a sufficient condition; also, it can be a demanding condition for some problems. See the "[[Courant–Friedrichs–Lewy condition#Implications of the CFL condition|Implications of the CFL condition]]" section of this article for a brief survey of these issues.</ref> It arises in the [[numerical analysis]] of [[explicit method|explicit]] time-marching 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 method|explicit]] time-marching [[computer simulation]]s, 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.<ref>See reference {{harvnb|Courant|Friedrichs|Lewy|1928}}. There exists also an [[English language|English]] [[translation]] of the 1928 [[German language|German]] original: see references {{harvnb|Courant|Friedrichs|Lewy|1956}} and {{harvnb|Courant|Friedrichs|Lewy|1967}}.</ref>
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| This condition is an example of [[temporal discretization|explicit time integration]] where the function that defines governing equation is evaluated at the current time.{{clarify|reason=What does {{''}}evaluated at the current time{{''}} mean? And what does {{''}}current time{{''}} refer to? What does it mean that the CFL condition is an {{''}}example{{''}} of explicit time integration? Isn't it rather be a restriction?|date=November 2013}}
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| ==Heuristic description==
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| 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 length,<ref>This situation commonly occurs when a [[hyperbolic partial differential equation|hyperbolic partial differential operator]] has been [[approximation theory|approximated]] by a [[Finite difference|finite difference equation]], which is then solved by [[numerical linear algebra]] methods.</ref> 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. 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.
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| ==The CFL condition==
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| In order to make a reasonably formally precise statement of the condition, it is necessary to define the following quantities
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| *''Spatial coordinate'': it is one of the [[coordinate]]s of the [[physical space]] in which the problem is posed.
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| *''Spatial dimension of the problem'': it is the number <math>n</math> of [[Dimension (mathematics)|spatial dimensions]] i.e. the number of spatial [[coordinate]]s of the [[physical space]] where the problem is posed. Typical values are <math>n=1</math>, <math>n=2</math> and <math>n=3</math>.
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| *''Time'': it is the [[coordinate]], acting as a [[parameter]], which describes the evolution of the system, distinct from the spatial coordinates.
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| The spatial coordinates and the time are supposed to be discrete-valued independent [[Variable (mathematics)|variables]], which are placed at regular distances called the ''interval length''<ref>This quantity is not necessarily the same for each spatial variable, as it is shown in the "[[Courant–Friedrichs–Lewy condition#The two and general ''n''–dimensional case|The two and general ''n''–dimensional case]]" section of this entry : it can be chosen in order to somewhat relax the condition.</ref> and the ''time step'', respectively. Using these names, the CFL condition relates the length of the time step to a function of the interval lengths of each spatial coordinate and of the maximum speed with which information can travel in the physical space.
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| Operatively, the CFL condition is commonly prescribed for those terms of the [[finite-difference approximation]] of general [[partial differential equation]]s which model the [[advection]] phenomenon.<ref>Precisely, this is the hyperbolic part of the PDE under analysis.</ref>
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| ===The one-dimensional case===
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| For one-dimensional case, the CFL has the following form:
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| : <math>C = \frac {u\,\Delta t} {\Delta x} \leq C_{max} </math>
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| where the [[dimensionless number]] is called the '''Courant number''',
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| *<math>u</math> is the [[Magnitude (mathematics)|magnitude]] of the velocity (whose [[Dimensional analysis#Definition|dimension]] is length/time)
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| *<math>\Delta t</math> is the time step (whose [[Dimensional analysis#Definition|dimension]] is time)
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| *<math>\Delta x</math> is the length interval (whose [[Dimensional analysis#Definition|dimension]] is length).
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| The value of <math>C_{max}</math> changes with the method used to solve the discretised equation, especially depending on whether the method is [[Explicit and implicit methods|explicit or implicit]]. If an explicit (time-marching) solver is used then typically <math>C_{max} = 1</math>. Implicit (matrix) solvers are usually less sensitive to numerical instability and so larger values of <math>C_{max}</math> may be tolerated.
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| ===The two and general ''n''-dimensional case===
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| In the [[Dimension (mathematics)|two-dimensional]] case, the CFL condition becomes
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| :<math>C = \frac {u_ x\,\Delta t}{\Delta x} + \frac {u_ y\,\Delta t}{\Delta y} \leq C_{max} </math>
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| with obvious meaning of the symbols involved. By analogy with the two-dimensional case, the general CFL condition for the <math>n</math>-dimensional case is the following one:
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| :<math>C = \Delta t \sum_{i=1}^n\frac{u_{x_i}}{\Delta x_i} \leq C_{max}. </math>
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| The interval length is not required to be the same for each spatial variable <math>\Delta x_i, i = 1, ..., n</math>. This "[[Degrees of freedom (physics and chemistry)|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.
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| ==Implications of the CFL condition==
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| ===The CFL condition is only a necessary one===
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| The CFL condition is a necessary condition, but may not be sufficient for the convergence of the [[finite-difference approximation]] of a given [[numerical method|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.
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| ===The CFL condition can be a very strong requirement===
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| The CFL condition can be a very limiting constraint on the time step <math>\Delta t</math>: for example, in the [[finite-difference approximation]] of certain fourth-order nonlinear partial differential equations, it can have the following form:{{fact|date=November 2013}}
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| : <math>\frac{\Delta t}{(\Delta x)^4} < C u</math>
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| meaning that a decrease in the length interval <math>\Delta x</math> requires a fourth order decrease in the time step <math>\Delta t</math> 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 method]]s.
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| == Notes ==
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| {{reflist|30em}}
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| == References ==
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| *{{Citation
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| | last = Courant
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| | first = R.
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| | author-link = Richard Courant
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| | last2 = Friedrichs
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| | first2 = K.
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| | author2-link = Kurt Otto Friedrichs
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| | last3 = Lewy
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| | first3 = H.
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| | author3-link = Hans Lewy
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| | title = Über die partiellen Differenzengleichungen der mathematischen Physik
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| | journal = [[Mathematische Annalen]]
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| | volume = 100
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| | issue = 1
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| | pages = 32–74
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| | year = 1928
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| | month =
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| | language = [[German language|German]]
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| | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002272636
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| | doi = 10.1007/BF01448839
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| | jfm = 54.0486.01
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| | mr = 1512478
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| |bibcode = 1928MatAn.100...32C }}.
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| *{{Citation
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| | last = Courant
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| | first = R.
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| | author-link = Richard Courant
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| | last2 = Friedrichs
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| | first2 = K.
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| | author2-link = Kurt Otto Friedrichs
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| | last3 = Lewy
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| | first3 = H.
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| | author3-link = Hans Lewy
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| | title = On the partial difference equations of mathematical physics
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| | place = New York
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| | publisher = AEC Computing and Applied Mathematics Centre – [[Courant Institute of Mathematical Sciences]]
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| | series = AEC Research and Development Report
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| | volume = NYO-7689
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| | origyear = 1928
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| | year = 1956
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| | month= September
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| | edition =
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| | pages = V + 76
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| | language =
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| | url = http://www.archive.org/details/onpartialdiffere00cour
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| | archiveurl = http://www.archive.org/stream/onpartialdiffere00cour#page/n0/mode/2up
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| | archivedate = October 23, 2008
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| | doi =
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| | id =
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| | isbn =
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| }}.: translated from the [[German language|German]] by Phyllis Fox. This is an earlier version of the paper {{harvnb|Courant|Friedrichs|Lewy|1967}}, circulated as a research report.
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| *{{Citation
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| | last = Courant
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| | first = R.
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| | author-link = Richard Courant
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| | last2 = Friedrichs
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| | first2 = K.
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| | author2-link = Kurt Otto Friedrichs
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| | last3 = Lewy
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| | first3 = H.
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| | author3-link = Hans Lewy
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| | title = On the partial difference equations of mathematical physics
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| | journal = [http://researchweb.watson.ibm.com/journal/rdindex.html IBM Journal of Research and Development]
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| | volume = 11
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| | issue = 2
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| | pages = 215–234
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| | origyear = 1928
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| | year = 1967
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| | month = March
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| | url = http://domino.research.ibm.com/tchjr/journalindex.nsf/a3807c5b4823c53f85256561006324be/769774a3c9f3685f85256bfa00683f8a!OpenDocument
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| | mr = 0213764
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| | zbl = 0145.40402
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| }}. A freely downlodable copy can be found [http://www.stanford.edu/class/cme324/classics/courant-friedrichs-lewy.pdf here].
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| == External links ==
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| *{{springer
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| | title=Courant–Friedrichs–Lewy condition
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| | id= C/c026760
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| | last= Bakhvalov
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| | first= N. S.
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| | author-link= Nikolai Sergeevich Bakhvalov
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| }}
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| *{{MathWorld |title=Courant-Friedrichs-Lewy Condition |id=Courant-Friedrichs-LewyCondition}}
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| {{DEFAULTSORT:Courant-Friedrichs-Lewy condition}}
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| [[Category:Numerical differential equations]]
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| [[Category:Computational fluid dynamics]]
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I am Oscar and I totally dig that title. I used to be unemployed but now I am a librarian and the wage has been truly fulfilling. One of the issues he loves most is ice skating but he is having difficulties to discover time for it. My family life in Minnesota and my family enjoys it.
My webpage; ftsacademy.com