finite difference wikipedia

7/29/2019 finite difference wikipedia

1/6

Finite difference method 1

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

Assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem,

where n! denotes the factorial of n, and Rn(x) is a remainder term, denoting the difference between the Taylor

polynomial of degree n and the original function. Again using the first derivative of the function fas an example, by

Taylor's theorem,

Setting, x0

=a and (x-a)=h we have,

Dividing across by h gives:

Solving for f'(a):

so that for sufficiently small,
http://en.wikipedia.org/w/index.php?title=Factorialhttp://en.wikipedia.org/w/index.php?title=Taylor%27s_theoremhttp://en.wikipedia.org/w/index.php?title=Finite_differencehttp://en.wikipedia.org/w/index.php?title=Differential_equationshttp://en.wikipedia.org/w/index.php?title=Differential_equationshttp://en.wikipedia.org/w/index.php?title=Numerical_methodshttp://en.wikipedia.org/w/index.php?title=Mathematics


2/6


Accuracy and order

The error in a method's solution is defined as the difference between its approximation and the exact analytical

solution. The two sources of error in finite difference methods are round-off error, the loss of precision due to

computer rounding of decimal quantities, and truncation error or discretization error, the difference between the

exact solution of the finite difference equation and the exact quantity assuming perfect arithmetic (that is, assuming

no round-off).

The finite difference method relies on discretizing

a function on a grid.

To use a finite difference method to attempt to solve (or, more

generally, approximate the solution to) a problem, one must first

discretize the problem's domain. This is usually done by dividing the

domain into a uniform grid (see image to the right). Note that this

means that finite-difference methods produce sets of discrete numerical

approximations to the derivative, often in a "time-stepping" manner.

An expression of general interest is the local truncation error of a

method. Typically expressed using Big-O notation, local truncation

error refers to the error from a single application of a method. That is,

it is the quantity if refers to the exact value and

to the numerical approximation. The remainder term of a Taylor

polynomial is convenient for analyzing the local truncation error.

Using the Lagrange form of the remainder from the Taylor polynomial

for , which is

, where

,

the dominant term of the local truncation error can be discovered. For example, again using the forward-difference

formula for the first derivative, knowing that ,

and with some algebraic manipulation, this leads to

and further noting that the quantity on the left is the approximation from the finite difference method and that the

quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation

error. A final expression of this example and its order is:

This means that, in this case, the local truncation error is proportional to the step size.
http://en.wikipedia.org/w/index.php?title=Big-O_notationhttp://en.wikipedia.org/w/index.php?title=Local_truncation_errorhttp://en.wikipedia.org/w/index.php?title=File%3AFinite_Differences.svghttp://en.wikipedia.org/w/index.php?title=Discretization_errorhttp://en.wikipedia.org/w/index.php?title=Truncation_errorhttp://en.wikipedia.org/w/index.php?title=Round-off_error


3/6


Example: ordinary differential equation

For example, consider the ordinary differential equation

The Euler method for solving this equation uses the finite difference quotient

to approximate the differential equation by first substituting in for u'(x) and applying a little algebra to get

The last equation is a finite-difference equation, and solving this equation gives an approximate solution to the

differential equation.

Example: The heat equation

Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions

(boundary condition)

(initial condition)

One way to numerically solve this equation is to approximate all the derivatives by finite differences. We partition

the domain in space using a mesh and in time using a mesh . We assume a uniform partition

both in space and in time, so the difference between two consecutive space points will be h and between two

consecutive time points will be k. The points

will represent the numerical approximation of

Explicit method

The stencil for the most common explicit methodfor the heat equation.

Using a forward difference at time and a second-order central

difference for the space derivative at position (FTCS) we get the

recurrence equation:

This is an explicit method for solving the one-dimensional heat

equation.

We can obtain from the other values this way:

where

So, with this recurrence relation, and knowing the values at time n, one can obtain the corresponding values at time

n+1. and must be replaced by the boundary conditions, in this example they are both 0.

This explicit method is known to be numerically stable and convergent whenever .[1]

The numerical errors

are proportional to the time step and the square of the space step:
http://en.wikipedia.org/w/index.php?title=Limit_of_a_sequencehttp://en.wikipedia.org/w/index.php?title=Numerically_stablehttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Explicit_methodhttp://en.wikipedia.org/w/index.php?title=FTCS_schemehttp://en.wikipedia.org/w/index.php?title=Central_differencehttp://en.wikipedia.org/w/index.php?title=Central_differencehttp://en.wikipedia.org/w/index.php?title=Forward_differencehttp://en.wikipedia.org/w/index.php?title=File%3AExplicit_method-stencil.svghttp://en.wikipedia.org/w/index.php?title=Stencil_%28numerical_analysis%29http://en.wikipedia.org/w/index.php?title=Dirichlet_boundary_conditionhttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Numerical_ordinary_differential_equations%23The_Euler_method


4/6


Implicit method

The implicit method stencil.

If we use the backward difference at time and a second-order

central difference for the space derivative at position (The

Backward Time, Centered Space Method "BTCS") we get the

recurrence equation:

This is an implicit method for solving the one-dimensional heat

equation.

We can obtain from solving a system of linear equations:

The scheme is always numerically stable and convergent but usually more numerically intensive than the explicit

method as it requires solving a system of numerical equations on each time step. The errors are linear over the time

step and quadratic over the space step.

CrankNicolson method

Finally if we use the central difference at time and a second-order central difference for the space derivative

at position ("CTCS") we get the recurrence equation:

This formula is known as the CrankNicolson method.

The CrankNicolson stencil.

We can obtain from solving a system of linear equations:

The scheme is always numerically stable and convergent but usually

more numerically intensive as it requires solving a system of numerical

equations on each time step. The errors are quadratic over both the

time step and the space step:

Usually the CrankNicolson scheme is the most accurate scheme for

small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement

and the least numerically intensive. The implicit scheme works the best for large time steps.

References

[1] Crank, J. The Mathematics of Diffusion. 2nd Edition, Oxford, 1975, p. 143.

K.W. Morton and D.F. Mayers,Numerical Solution of Partial Differential Equations, An Introduction.

Cambridge University Press, 2005.

Oliver Rbenknig, The Finite Difference Method (FDM) - An introduction (http://www.imtek.de/simulation/

mathematica/IMSweb/imsTOC/Lectures and Tips/Simulation I/FDM_introDocu.html), (2006) Albert

Ludwigs University of Freiburg

Autar Kaw and E. Eric Kalu,Numerical Methods with Applications, (2008) (http://www.autarkaw.com/books/

numericalmethods/

index.html)
http://www.autarkaw.com/books/numericalmethods/index.htmlhttp://www.autarkaw.com/books/numericalmethods/index.htmlhttp://en.wikipedia.org/w/index.php?title=Albert_Ludwigs_University_of_Freiburghttp://en.wikipedia.org/w/index.php?title=Albert_Ludwigs_University_of_Freiburghttp://www.imtek.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FDM_introDocu.htmlhttp://www.imtek.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FDM_introDocu.htmlhttp://en.wikipedia.org/w/index.php?title=Numerically_stablehttp://en.wikipedia.org/w/index.php?title=File%3ACrank-Nicolson-stencil.svghttp://en.wikipedia.org/w/index.php?title=Crank%E2%80%93Nicolson_methodhttp://en.wikipedia.org/w/index.php?title=Numerically_stablehttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Implicit_methodhttp://en.wikipedia.org/w/index.php?title=Backward_differencehttp://en.wikipedia.org/w/index.php?title=File%3AImplicit_method-stencil.svg


5/6


External links

List of Internet Resources for the Finite Difference Method for PDEs (http://math.fullerton.edu/mathews/

n2003/finitediffpde/FiniteDifferencePDEBib/Links/FiniteDifferencePDEBib_lnk_1.html)

Finite Difference Method of Solving ODEs (Boundary Value Problems) Notes, PPT, Maple, Mathcad, Matlab,

Mathematica (http://numericalmethods.eng.usf.edu/topics/finite_difference_method.html)

Lecture Notes (http://ltl.iams.sinica. edu.tw/document/training_lectures/2006/SH_Chen/

Finite_Difference_Methods.pdf) Shih-Hung Chen, National Central University

Randall J. LeVeque,Finite Difference Methods for Ordinary and Partial Differential Equations (http://faculty.

washington.edu/rjl/fdmbook/), SIAM, 2007.

Finite Difference Method (http://www.adeptscience.co.uk/products/mathsim/maple/powertools/des/unit18.

html#MapleAutoBookmark4)

Finite Difference Method for Boundary Value Problems (http://math.fullerton.edu/mathews/n2003/

FiniteDifferenceMod.html)

Finite Difference Methodology in Materials Science (http://www.composite-agency.com/product.htm)
http://www.composite-agency.com/product.htmhttp://math.fullerton.edu/mathews/n2003/FiniteDifferenceMod.htmlhttp://math.fullerton.edu/mathews/n2003/FiniteDifferenceMod.htmlhttp://www.adeptscience.co.uk/products/mathsim/maple/powertools/des/unit18.html#MapleAutoBookmark4http://www.adeptscience.co.uk/products/mathsim/maple/powertools/des/unit18.html#MapleAutoBookmark4http://faculty.washington.edu/rjl/fdmbook/http://faculty.washington.edu/rjl/fdmbook/http://en.wikipedia.org/w/index.php?title=Randall_J._LeVequehttp://en.wikipedia.org/w/index.php?title=National_Central_Universityhttp://ltl.iams.sinica.edu.tw/document/training_lectures/2006/SH_Chen/Finite_Difference_Methods.pdfhttp://ltl.iams.sinica.edu.tw/document/training_lectures/2006/SH_Chen/Finite_Difference_Methods.pdfhttp://numericalmethods.eng.usf.edu/topics/finite_difference_method.htmlhttp://math.fullerton.edu/mathews/n2003/finitediffpde/FiniteDifferencePDEBib/Links/FiniteDifferencePDEBib_lnk_1.htmlhttp://math.fullerton.edu/mathews/n2003/finitediffpde/FiniteDifferencePDEBib/Links/FiniteDifferencePDEBib_lnk_1.html


6/6

Article Sources and Contributors 6

Article Sources and ContributorsFinite difference method Source: http://en.wikipedia.org/w/index.php?oldid=529066075 Contributors: Adelpine, Ajgorhoe, Andreasmperu, Ary29, AtomicDragon, Ben pcc, Berland, Charvest,

Ched, Dlandau, Duplico, EconoPhysicist, Fintor, J04n, JRSpriggs, Jhbayless, Jitse Niesen, Jmath666, Larryisgood, Leperous, Mhym, Michael Hardy, Mintz l, Mwtoews, Nbarth, Neves100,

Nicoguaro, Okypris, Oleg Alexandrov, Paul August, Rbonvall, Rudolf.hellmuth, Salih, Smcreator, Superale85, User A1, Vilietha, Wallpaperdesktop, , 73 anonymous edits

Image Sources, Licenses and ContributorsFile:Finite Differences.svg Source: http://en.wikipedia.org/w/index.php?title=File:Finite_Differences.svg License: Public Domain Contributors: User:Mintz l

File:Explicit method-stencil.svg Source: http://en.wikipedia.org/w/index.php?title=File:Explicit_method-stencil.svg License: Public Domain Contributors: derivative work: Mintz l (talk)

Crank-Nicolson-stencil.svg: Original uploader was Berland at en.wikipedia

File:Implicit method-stencil.svg Source: http://en.wikipedia.org/w/index.php?title=File:Implicit_method-stencil.svg License: Public Domain Contributors: derivative work: Mintz l (talk)

Crank-Nicolson-stencil.svg: Original uploader was Berland at en.wikipedia

File:Crank-Nicolson-stencil.svg Source: http://en.wikipedia.org/w/index.php?title=File:Crank-Nicolson-stencil.svg License: Public Domain Contributors: Original uploader was Berland at

en.wikipedia

License

Creative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/

Documents

finite difference wikipedia