Call
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 {\displaystyle u} , the exact solution to a differential equation in an appropriate normed space {\displaystyle }. Consider a numerical approximation u h {\displaystyle u_{h}} , where h {\displaystyle 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 {\displaystyle u_{h}} is said to be n {\displaystyle n} th-order accurate if the error E := | | u − u h | | {\displaystyle E:=||u-u_{h}||} is proportional to the step-size h {\displaystyle h} to the n {\displaystyle n} th power:
where the constant C {\displaystyle C} is independent of h {\displaystyle h} and usually depends on the solution u {\displaystyle u}. Using the big O notation an n {\displaystyle n} 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 h {\displaystyle h}.Partial differential equations which vary over both time and space are said to be accurate to order n {\displaystyle n} in time and to order m {\displaystyle m} in space.
Talk Doctor Online in Bissoy App