In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. 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 (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra 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.

For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as $${\displaystyle f(x_{0}+h)=f(x_{0})+{\frac {f'(x_{0})}{1!}}h+{\frac {f^{(2)}(x_{0})}{2!}}h^{2}+\cdots +{\frac {f^{(n)}(x_{0})}{n!}}h^{n}+R_{n}(x),}$$

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.

Following is the process to derive an approximation for the first derivative of the function f by first truncating the Taylor polynomial plus remainder: $${\displaystyle f(x_{0}+h)=f(x_{0})+f'(x_{0})h+R_{1}(x).}$$ Dividing across by h gives: $${\displaystyle {f(x_{0}+h) \over h}={f(x_{0}) \over h}+f'(x_{0})+{R_{1}(x) \over h}}$$ Solving for ${\displaystyle f'(x_{0})}$: $${\displaystyle f'(x_{0})={f(x_{0}+h)-f(x_{0}) \over h}-{R_{1}(x) \over h}.}$$

Assuming that ${\displaystyle R_{1}(x)}$ is sufficiently small, the approximation of the first derivative of f is: $${\displaystyle f'(x_{0})\approx {f(x_{0}+h)-f(x_{0}) \over h}.}$$

This is similar to the definition of derivative, which is: $${\displaystyle f'(x_{0})=\lim _{h\to 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}.}$$ except for the limit towards zero (the method is named after this).

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 (no round-off).