Finite difference method
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In mathematics, finite-difference 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. FDMs convert a linear (non-linear) ODE/PDE into a system of linear (non-linear) equations, which can then be solved by matrix algebra techniques. The reduction of the differential equation to a system of algebraic equations makes the problem of finding the solution to a given ODE ideally suited to modern computers, hence the widespread use of FDMs in modern numerical analysis.
Derivation from Taylor's polynomialEdit
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. We will derive an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial:
Setting, x0=a we have,
Dividing across by h gives:
Solving for f'(a):
Assuming that is sufficiently small, the approximation of the first derivative of "f" is:
Accuracy and orderEdit
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, 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 original differential equation and the exact quantity assuming perfect arithmetic (that is, assuming 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 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 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.
Example: ordinary differential equationEdit
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 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 equationEdit
- (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
We can obtain from the other values this way:
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.
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:
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:
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 Crank–Nicolson method.
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 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.
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 operatorEdit
The (continuous) Laplace operator 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
The 2D case shows all the characteristics of the more general nD case. Each second partial derivative needs to be approximated similar to the 1D case
which is usually given by the following stencil
Consistency of the above-mentioned approximation can be shown for highly regular functions, such as . The statement is
To proof this one needs to substitute Taylor Series expansions up to order 3 into the discrete Laplace operator.
Similar to continuous subharmonic functions one can define subharmonic functions for finite-difference approximations
One can define a general stencil 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.
For a (discrete) subharmonic function the following holds
where are discretizations of the continuous domain , respectively the boundary .
A similar maximum principle also holds for the continuous case.
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- Finite difference coefficients
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- Discrete Poisson equation
- Discrete Laplace operator
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