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# 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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