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orderofaccuracy

order of accuracy

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There are 15 letters in ORDEROFACCURACY ( A1C3D2E1F4O1R1U1Y4 )

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Definitions of order of accuracy in various dictionaries:

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 solu...

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Order of accuracy description
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
*
*
*
* (
* V
* ,
*
* |
*
*
* |
*
*
*
* |
*
*
* |
*
* )
*
*
* {\displaystyle (V,||\ ||)}
* . Consider a numerical approximation
*
*
*
*
* u
*
* h
*
*
*
*
* {\displaystyle u_{h}}
* , where
*
*
*
* h
*
*
* {\displaystyle h}
* is a parameter characterizing the approximation, 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
* (
* h
* )
* :=
*
* |
*
*
* |
*
* u
* −
*
* u
*
* h
*
*
*
* |
*
*
* |
*
*
*
* {\displaystyle E(h):=||u-u_{h}||}
* is proportional to the step-size
*
*
*
* h
*
*
* {\displaystyle h}
* to the
*
*
*
* n
*
*
* {\displaystyle n}
* th power;*
*
*
* E
* (
* h
* )
* =
*
* |
*
*
* |
*
* u
* −
*
* u
*
* h
*
*
*
* |
*
*
* |
*
* ≤
* C
*
* h
*
* n
*
*
*
*
* {\displaystyle E(h)=||u-u_{h}||\leq Ch^{n}}
* Where the constant
*
*
*
* C
*
*
* {\displaystyle C}
* is independent of 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
*
*
*
*
*
* |
*
*
* |
*
* u
* −
*
* u
*
* h
*
*
*
* |
*
*
* |
*
* =
* O
* (
*
* h
*
* n
*
*
* )
*
*
* {\displaystyle ||u-u_{h}||=O(h^{n})}
* 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...
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