Courant–Friedrichs–Lewy condition

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Courant–Friedrichs–Lewy condition (CFL condition) is a necessary condition for convergence while solving certain partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s (usually hyperbolic PDE

Hyperbolic partial differential equation

In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...

s) numerically by the method of finite differences

Finite difference method

In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...

. 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 simulation

Computer simulation

A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s, otherwise the simulation will produce wildly incorrect results. The condition is named after Richard Courant

Richard Courant

Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...

, Kurt Friedrichs, and Hans Lewy

Hans Lewy

Hans Lewy was an American mathematician, known for his work on partial differential equations and on the theory of functions of several complex variables....

who described it in their 1928 paper.

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

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

at discrete time steps of equal length, 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 of spatial dimensions i.e. the number of spatial coordinates of the physical space where the problem is posed. Typical values are , and .
Time: it is the coordinate, acting as a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

, 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

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

, whose minimal steps are called respectively the interval length 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 equation

Partial differential equation

s which model the advection

Advection

Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...

phenomenon.

The one–dimensional case

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

where

is the velocity (whose dimension is Length/Time)
is the time step (whose dimension is Time)
is the length interval (whose dimension is Length),
is a dimensionless constant which depends only on the particular equation to be solved.

The dimensionless number

is called the Courant number.

The two and general n–dimensional case

In the two–dimensional case, the CFL condition becomes

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

–dimensional case is the following one

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

= 1 ,...,

. 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.

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

: for example, in the finite-difference approximation of certain fourth-order nonlinear partial differential equations, it can have the following form

meaning that a decrease in the length interval

requires a fourth order decrease in the time step

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, a modern dynamical systems approaches to modeling, based upon center manifold

Center manifold

In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...