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,|| ||). Consider a numerical approximation uh, 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 uh is said to be nth-order accurate if the error E(h):=||u−uh|| is proportional to the step-size h to the nth power:[1]

E(h)=||u−uh||≤Chn

where the constant C is independent of h and usually depends on the solution u.[2] Using the big O notation an nth-order accurate numerical method is notated as

||u−uh||=O(hn)

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]

## References

1. ↑ LeVeque, Randall J (2006). Finite Difference Methods for Differential Equations. University of Washington. pp. 3–5.
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.
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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