Order of accuracy explained

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]

Notes and References

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