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 , the exact solution to a differential equation in an appropriate normed space . Consider a numerical approximation , where 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 is said to be th-order accurate if the error, is proportional to the step-size to the th power;[1]

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

This definition is strictly depended 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 . Partial differential equations which vary over both time and space are said to be accurate to order in time and to order 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.
  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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