In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, finite-difference methods (FDM) are a class of numerical techniques for solving
differential equations by approximating
derivatives with
finite differences
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
. Both the spatial domain and time domain (if applicable) are
discretized, or broken into a finite number of intervals, and the values of the solution at the end points of the intervals are approximated by solving algebraic equations containing finite differences and values from nearby points.
Finite difference methods convert
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
(ODE) or
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
(PDE), which may be
nonlinear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
, into a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables.
For example,
: \begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of th ...
that can be solved by
matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alterna ...
techniques. Modern computers can perform these
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
computations efficiently, and this, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis.
[
Today, FDMs are one of the most common approaches to the numerical solution of PDE, along with finite element methods.]
Derive difference quotient from Taylor's polynomial
For a ''n''-times differentiable function, by Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...
the Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion is given as
Where ''n''! denotes the factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
of ''n'', and ''R''''n''(''x'') is a remainder term, denoting the difference between the Taylor polynomial of degree ''n'' and the original function.
Following is the process to derive an approximation for the first derivative of the function ''f'' by first truncating the Taylor polynomial plus remainder:
Dividing across by ''h'' gives:
Solving for :
Assuming that is sufficiently small, the approximation of the first derivative of ''f'' is:
This is similar to the definition of derivative, which is:
except for the limit towards zero (the method is named after this).
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 round-off error
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Roun ...
, the loss of precision due to computer rounding of decimal quantities, and truncation error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. The term truncation comes from the fact that these simplifications often involve the truncation of an infinite series expa ...
or discretization error, the difference between the exact solution of the original differential equation and the exact quantity assuming perfect arithmetic (no round-off).
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). 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
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Pau ...
, 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 the Taylor polynomial can be used to analyze local truncation error. Using the Lagrange form of the remainder from the Taylor polynomial for , which is
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:
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. Therefore, a reasonable balance between data quality and simulation duration is necessary for practical usage. Large time steps are useful for increasing simulation speed in practice. However, time steps which are too large may create instabilities and affect the data quality.
The von Neumann and Courant-Friedrichs-Lewy criteria are often evaluated to determine the numerical model stability.
Example: ordinary differential equation
For example, consider the ordinary differential equation
The Euler method
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical analysis, numerical procedure for solving ordinary differential equations (ODEs) with a given Initial value problem, in ...
for solving this equation uses the finite difference quotient
to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) 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 mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
in one dimension, with homogeneous Dirichlet boundary condition
In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...
s
One way to numerically solve this equation is to approximate all the derivatives by finite differences. First partition the domain in space using a mesh and in time using a mesh . 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
Using a forward difference at time and a second-order central difference
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly ...
for the space derivative at position ( FTCS) gives the recurrence equation:
This is an explicit method for solving the one-dimensional heat equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
.
One 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 . The numerical errors are proportional to the time step and the square of the space step:
Implicit method
Using the backward difference at time and a second-order central difference for the space derivative at position (The Backward Time, Centered Space Method "BTCS") gives the recurrence equation:
This is an implicit method for solving the one-dimensional heat equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
.
One 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:
Crank–Nicolson method
Finally, using the central difference at time and a second-order central difference for the space derivative at position ("CTCS") gives the recurrence equation:
This formula is known as the Crank–Nicolson method.
One 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:
Comparison
To summarize, usually the Crank–Nicolson scheme is the most accurate scheme for small time steps. For larger time steps, the implicit scheme works better since it is less computationally demanding. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive.
Here is an example. The figures below present the solutions given by the above methods to approximate the heat equation
with the boundary condition
The exact solution is
Example: The Laplace operator
The (continuous) Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
in -dimensions is given by .
The discrete Laplace operator depends on the dimension .
In 1D the Laplace operator is approximated as
This approximation is usually expressed via the following stencil
Stencilling produces an image or pattern on a surface by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object. The holes allow the pigment to reach only some parts of the surface creatin ...
and which represents a symmetric, tridiagonal matrix.
For an equidistant grid one gets a Toeplitz matrix.
The 2D case shows all the characteristics of the more general n-dimensional case. Each second partial derivative needs to be approximated similar to the 1D case
which is usually given by the following stencil
Stencilling produces an image or pattern on a surface by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object. The holes allow the pigment to reach only some parts of the surface creatin ...
Consistency
Consistency of the above-mentioned approximation can be shown for highly regular functions, such as .
The statement is
To prove this, one needs to substitute Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansions up to order 3 into the discrete Laplace operator.
Properties
Subharmonic
Similar to continuous subharmonic functions one can define ''subharmonic functions'' for finite-difference approximations
Mean value
One can define a general stencil
Stencilling produces an image or pattern on a surface by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object. The holes allow the pigment to reach only some parts of the surface creatin ...
of ''positive type'' via
If is (discrete) subharmonic then the following'' mean value property'' holds
where the approximation is evaluated on points of the grid, and the stencil is assumed to be of positive type.
A similar mean value property also holds for the continuous case.
Maximum principle
For a (discrete) subharmonic function the following holds
where are discretizations of the continuous domain , respectively the boundary .
A similar maximum principle
In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle i ...
also holds for the continuous case.
The SBP-SAT method
The SBP-SAT (''summation by parts - simultaneous approximation term'') method is a stable and accurate technique for discretizing and imposing boundary conditions of a well-posed partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
using high order finite differences.
The method is based on finite differences where the differentiation operators exhibit summation-by-parts properties. Typically, these operators consist of differentiation matrices with central difference stencils in the interior with carefully chosen one-sided boundary stencils designed to mimic integration-by-parts in the discrete setting. Using the SAT technique, the boundary conditions of the PDE are imposed weakly, where the boundary values are "pulled" towards the desired conditions rather than exactly fulfilled. If the tuning parameters (inherent to the SAT technique) are chosen properly, the resulting system of ODE's will exhibit similar energy behavior as the continuous PDE, i.e. the system has no non-physical energy growth. This guarantees stability if an integration scheme with a stability region that includes parts of the imaginary axis, such as the fourth order Runge-Kutta method, is used. This makes the SAT technique an attractive method of imposing boundary conditions for higher order finite difference methods, in contrast to for example the injection method, which typically will not be stable if high order differentiation operators are used.
See also
* Finite element method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...
* Finite difference
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
* Finite difference time domain
* Infinite difference method
* Stencil (numerical analysis)
In mathematics, especially the areas of numerical analysis concentrating on the numerical partial differential equations, numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the ...
* Finite difference coefficients
* Five-point stencil
In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations t ...
* Lax–Richtmyer theorem
* Finite difference methods for option pricing Finite difference methods for option pricing are Numerical analysis, numerical methods used in mathematical finance for the valuation of Option (finance), options. Finite difference methods were first applied to Valuation of options, option pricing ...
* Upwind differencing scheme for convection
* Central differencing scheme
* Discrete Poisson equation
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical an ...
* Discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a Graph (discrete mathematics), graph or a lattice (group), discrete grid. For the case of a finite-dimensional graph ...
References
Further reading
* 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
Contains a brief, engineering-oriented introduction to FDM (for ODEs) i
Chapter 08.07
*
*
* .
* Randall J. LeVeque,
Finite Difference Methods for Ordinary and Partial Differential Equations
', SIAM, 2007.
* Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov(Eds): "Exact Finite-Difference Schemes", De Gruyter (2016). DOI: https://doi.org/10.1515/9783110491326 .
* Mikhail Shashkov: ''Conservative Finite-Difference Methods on General Grids'', CRC Press, ISBN 0-8493-7375-1 (1996).
{{DEFAULTSORT:Finite Difference Method
Finite differences
Numerical differential equations