Order of accuracy

Innumerical analysis,order of accuracyquantifies therate of convergenceof a numerical approximation of adifferential equationto the exact solution. Consider$u$,the exact solution to a differential equation in an appropriatenormed 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 afinite element method. The numerical solution$u_{h}$is said to be$n$th-order accurateif 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 thebig O notationan$n$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 equationswhich vary over both time and space are said to be accurate to order$n$in time and to order$m$in space.