# discretization

**discretization** The process of replacing a problem defined on a continuum, say an interval [0,1], by an approximating problem on a finite set of points, say *nh*, *n* = 0,1,2,…,*N*, where *h* = 1/*N*

Examples arise in many branches of numerical analysis, principally ordinary and partial differential equations where the finite-difference method and the finite-element method are common forms of discretization. For the ordinary differential equation *y′* = *f*(*x*,*y*), 0←*x*←1, *y*(0) = *y*_{0},

a simple discretization is given by *Euler's method*: (1/*h*)(*y _{n}*

_{+1}–

*y*) =

_{n}*f*(

*x*,

_{n}*y*)

_{n}where

*x*=

_{n}*hn*,

*n*= 0,1,…,

*N*,

*h*= 1/

*N*

and

*y*denotes the approximation to the true solution

_{n}*y*(

*x*) at the point

*x*. See also discretization error.

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