In mathematics, finitedifference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives. FDMs are thus discretization methods.
Today, FDMs are the dominant approach to numerical solutions of partial differential equations.^{[1]}
Derivation from Taylor's polynomial
First, assuming the function whose derivatives are to be approximated is properlybehaved, by Taylor's theorem, we can create a Taylor Series expansion

f(x_0 + h) = f(x_0) + \frac{f'(x_0)}{1!}h + \frac{f^{(2)}(x_0)}{2!}h^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}h^n + R_n(x),
where n! denotes the factorial of n, and R_{n}(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. We will derive an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial:

f(x_0 + h) = f(x_0) + f'(x_0)h + R_1(x),
Setting, x_{0}=a we have,

f(a+h) = f(a) + f'(a)h + R_1(x),
Dividing across by h gives:

{f(a+h)\over h} = {f(a)\over h} + f'(a)+{R_1(x)\over h}
Solving for f'(a):

f'(a) = {f(a+h)f(a)\over h}  {R_1(x)\over h}
Assuming that R_1(x) is sufficiently small, the approximation of the first derivative of "f" is:

f'(a)\approx {f(a+h)f(a)\over h}.
Accuracy and order
The error in a method's solution is defined as the difference between the approximation and the exact analytical solution. The two sources of error in finite difference methods are roundoff 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 roundoff).
The finite difference method relies on discretizing a function on a grid.
To use a finite difference method to 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 finitedifference methods produce sets of discrete numerical approximations to the derivative, often in a "timestepping" manner.
An expression of general interest is the local truncation error of a method. Typically expressed using BigO notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity f'(x_i)  f'_i if f'(x_i) refers to the exact value and f'_i 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 f(x_0 + h), which is

R_n(x_0 + h) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (h)^{n+1} , where x_0 < \xi < x_0 + h,
the dominant term of the local truncation error can be discovered. For example, again using the forwarddifference formula for the first derivative, knowing that f(x_i)=f(x_0+i h),

f(x_0 + i h) = f(x_0) + f'(x_0)i h + \frac{f''(\xi)}{2!} (i h)^{2},
and with some algebraic manipulation, this leads to

\frac{f(x_0 + i h)  f(x_0)}{i h} = f'(x_0) + \frac{f''(\xi)}{2!} i h,
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:

\frac{f(x_0 + i h)  f(x_0)}{i h} = f'(x_0) + O(h).
This means that, in this case, the local truncation error is proportional to the step sizes. The quality and duration of simulated FDM solution depends on the discretization equation selection and the step sizes (time and space steps). The data quality and simulation duration increase significantly with smaller step size.^{[2]} Therefore, a reasonable balance between data quality and simulation duration is necessary for practical usage. Large time steps are favourable to increase simulation speed in many practice, however too large time steps may create instabilities and affecting the data quality.^{[3]}^{[4]}
The von Neumann method (Fourier stability analysis) is usually applied to determine the numerical model stability.^{[3]}^{[4]}^{[5]}^{[6]}
Example: ordinary differential equation
For example, consider the ordinary differential equation

u'(x) = 3u(x) + 2. \,
The Euler method for solving this equation uses the finite difference quotient

\frac{u(x+h)  u(x)}{h} \approx u'(x)
to approximate the differential equation by first substituting in for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get

u(x+h) = u(x) + h(3u(x)+2). \,
The last equation is a finitedifference 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

U_t=U_{xx} \,

U(0,t)=U(1,t)=0 \, (boundary condition)

U(x,0) =U_0(x) \, (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 x_0, ..., x_J and in time using a mesh t_0, ...., t_N . 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

u(x_j,t_n) = u_{j}^n
will represent the numerical approximation of u(x_j, t_n).
Explicit method
The
stencil for the most common explicit method for the heat equation.
Using a forward difference at time t_n and a secondorder central difference for the space derivative at position x_j (FTCS) we get the recurrence equation:

\frac{u_{j}^{n+1}  u_{j}^{n}}{k} = \frac{u_{j+1}^n  2u_{j}^n + u_{j1}^n}{h^2}. \,
This is an explicit method for solving the onedimensional heat equation.
We can obtain u_j^{n+1} from the other values this way:

u_{j}^{n+1} = (12r)u_{j}^{n} + ru_{j1}^{n} + ru_{j+1}^{n}
where r=k/h^2.
So, with this recurrence relation, and knowing the values at time n, one can obtain the corresponding values at time n+1. u_0^n and u_J^n 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 r\le 1/2 .^{[7]} The numerical errors are proportional to the time step and the square of the space step:

\Delta u = O(k)+O(h^2) \,
Implicit method
The implicit method stencil.
If we use the backward difference at time t_{n+1} and a secondorder central difference for the space derivative at position x_j (The Backward Time, Centered Space Method "BTCS") we get the recurrence equation:

\frac{u_{j}^{n+1}  u_{j}^{n}}{k} =\frac{u_{j+1}^{n+1}  2u_{j}^{n+1} + u_{j1}^{n+1}}{h^2}. \,
This is an implicit method for solving the onedimensional heat equation.
We can obtain u_j^{n+1} from solving a system of linear equations:

(1+2r)u_j^{n+1}  ru_{j1}^{n+1}  ru_{j+1}^{n+1}= u_{j}^{n}
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:

\Delta u = O(k)+O(h^2). \,
Crank–Nicolson method
Finally if we use the central difference at time t_{n+1/2} and a secondorder central difference for the space derivative at position x_j ("CTCS") we get the recurrence equation:

\frac{u_j^{n+1}  u_j^{n}}{k} = \frac{1}{2} \left(\frac{u_{j+1}^{n+1}  2u_j^{n+1} + u_{j1}^{n+1}}{h^2}+\frac{u_{j+1}^{n}  2u_j^{n} + u_{j1}^{n}}{h^2}\right).\,
This formula is known as the Crank–Nicolson method.
The Crank–Nicolson stencil.
We can obtain u_j^{n+1} from solving a system of linear equations:

(2+2r)u_j^{n+1}  ru_{j1}^{n+1}  ru_{j+1}^{n+1}= (22r)u_j^n + ru_{j1}^n + ru_{j+1}^n
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:

\Delta u = O(k^2)+O(h^2). \,
Usually the Crank–Nicolson 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.
See also
References

^

^

^ ^{a} ^{b}

^ ^{a} ^{b}

^

^

^ 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.

Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008) [1]. Contains a brief, engineeringoriented introduction to FDM (for ODEs) in Chapter 08.07.



.

Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
External links

List of Internet Resources for the Finite Difference Method for PDEs
Various lectures and lecture notes

FiniteDifference Method in Electromagnetics (see and listen to lecture 9)

Lecture Notes ShihHung Chen, National Central University

Finite Difference Method for Boundary Value Problems

Numerical Methods for timedependent Partial Differential Equations
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.