Order of accuracy

In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. A numerical solution to a differential equation is said to be $n$th-order accurate if the error, $E$, is proportional to the step-size $h$ to the $n$th power;^[1]

$E(h) = Ch^n$

The size of the error of a first-order accurate approximation is directly proportional to $h$. In big O notation, an $n$th-order accurate numerical method is notated as $O(h^n)$. 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.^[2]

References

1. LeVeque, Randall J (2006). Finite Difference Methods for Differential Equations. University of Washington. pp. 3–5.
2. Strikwerda, John C (2004). Finite Difference Schemes and Partial Differential Equations (2 ed.). pp. 62–66. ISBN 978-0-898716-39-9.