Order of accuracy

Last updated May 08, 2023

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 $(V,||\ ||)$ . 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:^[1]

E(h)=||u-u_{h}||\leq Ch^{n}

where the constant $C$ is independent of $h$ and usually depends on the solution $u$ .^[2] 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.^[3]

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References

↑ LeVeque, Randall J (2006). Finite Difference Methods for Differential Equations. University of Washington. pp. 3–5. CiteSeerX 10.1.1.111.1693 .
↑ Ciarliet, Philippe J (1978). The Finite Element Method for Elliptic Problems. Elsevier. pp. 105–106. doi:10.1137/1.9780898719208. ISBN 978-0-89871-514-9.
↑ Strikwerda, John C (2004). Finite Difference Schemes and Partial Differential Equations (2 ed.). pp. 62–66. ISBN 978-0-898716-39-9.

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[LeVeque-1] LeVeque, Randall J (2006). Finite Difference Methods for Differential Equations. University of Washington. pp. 3–5. CiteSeerX 10.1.1.111.1693 .

[2] Ciarliet, Philippe J (1978). The Finite Element Method for Elliptic Problems. Elsevier. pp. 105–106. doi:10.1137/1.9780898719208. ISBN 978-0-89871-514-9.

[Strikwerda-3] Strikwerda, John C (2004). Finite Difference Schemes and Partial Differential Equations (2 ed.). pp. 62–66. ISBN 978-0-898716-39-9.

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