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 $$u$$, 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:


 * $$ E(h) = ||u-u_h|| \leq Ch^n $$

where the constant $$C$$ is independent of $$h$$ and usually depends on the solution $$u$$. 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.