Finite difference method

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.

Today, FDMs are the dominant approach to numerical solutions of partial differential equations.

First, assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem, we can create a Taylor Series expansion

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:

Setting, x₀=a we have,

Dividing across by h gives:

Solving for f'(a):

Assuming that ${\displaystyle R_{1}(x)}$ is sufficiently small, the approximation of the first derivative of "f" is:

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).

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