order of accuracy

{{Short description|Term in numerical analysis}}

In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution.

Consider $u$, the exact solution to a differential equation in an appropriate normed space $(V,||\backslash \; ||)$. Consider a numerical approximation $u\_h$, where $h$ is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method.

The numerical solution $u\_h$ is said to be $n$th-order accurate if the error $E(h):=\; ||u-u\_h||$ is proportional to the step-size $h$ to the $n$th power:{{cite book|last=LeVeque|first=Randall J|title=Finite Difference Methods for Differential Equations|year=2006|publisher=University of Washington|pages=3–5|citeseerx=10.1.1.111.1693}}

:$E(h)\; =\; ||u-u\_h||\; \backslash leq\; Ch^n$

where the constant $C$ is independent of $h$ and usually depends on the solution $u$.{{cite book|last=Ciarliet|first=Philippe J|title=The Finite Element Method for Elliptic Problems|year=1978|publisher=Elsevier|pages=105–106|doi=10.1137/1.9780898719208|isbn=978-0-89871-514-9}} Using the big O notation an $n$th-order accurate numerical method is notated as

:$||u-u\_h||\; =\; O(h^n)$

This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, all numerical errors correctly.

The size of the error of a first-order accurate approximation is directly proportional to $h$.

Partial differential equations which vary over both time and space are said to be accurate to order $n$ in time and to order $m$ in space.{{cite book|last=Strikwerda|first=John C|title=Finite Difference Schemes and Partial Differential Equations|edition=2|year=2004|isbn=978-0-898716-39-9|pages=62–66}}