August 8, 2022

Finite-difference methods (FDM), or simply difference methods, in numerical analysis are difference equations obtained by replacing differentiation by finite-difference approximations (difference quotients) to solve differential equations. It is a numerical solution method that uses a discretization method called approximation. It is said to have been invented by Euler in the 18th century. Today, FDM is the dominant method for numerical solution of partial differential equations.

Accuracy and error

Solution error is defined as the difference between the true analytical solution and the approximate solution. The sources of error in finite difference methods are rounding errors and truncation or discretization errors. In order to use the finite difference method to approximate the solution to a problem, the problem domain must first be discretized. This is usually done by dividing the area into a uniform grid. Note that this means that finite difference methods often provide a set of discrete numerical approximations to the derivative in a "time-step" fashion. f ( x i ) f ( x 0 + i h ) {\displaystyle f(x_{i})f(x_{0}+ih)} Of general interest is the local truncation error, which is typically expressed in O-notation. The local truncation error refers to the error at each point, the difference between the true value f'(xi) and the approximate value f'i f ′ ( x i ) − f i ′ {\displaystyle f'(x_{i})-f'_{i}} is. To evaluate this error, it is convenient to look at the remainder term of the Taylor expansion. Lagrangian remainder term of the Taylor expansion for the expression f(x0 + h) R. n ( x 0 + h ) f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( h ) n + 1 ( x 0 < ξ < x 0 + h ) {\displaystyle R_{n}(x_{0}+h){\frac {f^{(n+1)}(\xi )}{(n+1)!}}(h)^{n+ 1}\quad(x_{0}<\xi