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. Consider [math]\displaystyle{ u }[/math], the exact solution to a differential equation in an appropriate normed space [math]\displaystyle{ (V,||\ ||) }[/math]. Consider a numerical approximation [math]\displaystyle{ u_h }[/math], where [math]\displaystyle{ h }[/math] 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 [math]\displaystyle{ u_h }[/math] is said to be [math]\displaystyle{ n }[/math]th-order accurate if the error [math]\displaystyle{ E(h):= ||u-u_h|| }[/math] is proportional to the step-size [math]\displaystyle{ h }[/math] to the [math]\displaystyle{ n }[/math]th power:[1]
- [math]\displaystyle{ E(h) = ||u-u_h|| \leq Ch^n }[/math]
where the constant [math]\displaystyle{ C }[/math] is independent of [math]\displaystyle{ h }[/math] and usually depends on the solution [math]\displaystyle{ u }[/math].[2] Using the big O notation an [math]\displaystyle{ n }[/math]th-order accurate numerical method is notated as
- [math]\displaystyle{ ||u-u_h|| = O(h^n) }[/math]
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 [math]\displaystyle{ h }[/math]. Partial differential equations which vary over both time and space are said to be accurate to order [math]\displaystyle{ n }[/math] in time and to order [math]\displaystyle{ m }[/math] in space.[3]
References
- ↑ LeVeque, Randall J (2006). Finite Difference Methods for Differential Equations. University of Washington. pp. 3–5.
- ↑ 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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