Discretization Error (truncation error) Definition/Meaning:
The
error in a numerical method that has been constructed by a process of
discretization.
It results from the discretization of a "continuous" problem, where it is
assumed that all arithmetic is done exactly, and is of fundamental importance in
methods for differential equations. A distinction is made between global and
local errors. For example, in Euler's method (see discretization) yn is the
approximation to the solution y(x) at xn. The global discretization for
truncation) error is given by
yn - y(xn)
(Some authors take the maximum of this expression over n = 0,1, ... ,N.) The
local discretization (or truncation) error is the amount by which the true
solution of the differential equation fails to satisfy the discretization
formula, i.e.
y(xn+1) - y(xn) - hƒ(xn,y(xn))
This quantity can be regarded as a first measure of
the accuracy of the formula. It can be estimated at each step of the integration
and provides a means of indirectly controlling the global discretization error.
These definitions can be extended to other methods and beyond.
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