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:
where the constant is independent of and usually depends on the solution. Using the big O notation an th-order accurate numerical method is notated as
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.
Partial [differential equations] which vary over both time and space are said to be accurate to order in time and to order in space.