In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
,
computational physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science ...
, and
simulation
A simulation is an imitative representation of a process or system that could exist in the real world. In this broad sense, simulation can often be used interchangeably with model. Sometimes a clear distinction between the two terms is made, in ...
, discretization error is the
error
An error (from the Latin , meaning 'to wander'Oxford English Dictionary, s.v. “error (n.), Etymology,” September 2023, .) is an inaccurate or incorrect action, thought, or judgement.
In statistics, "error" refers to the difference between t ...
resulting from the fact that a
function of a
continuous variable is represented in the computer by a finite number of evaluations, for example, on a
lattice. Discretization error can usually be reduced by using a more finely spaced lattice, with an increased
computational cost
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.
Mechanical or electronic devices (or, historic ...
.
Examples
Discretization error is the principal source of error in methods of
finite difference
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
s and the
pseudo-spectral method of computational physics.
When we define the derivative of
as
or
, where
is a finitely small number, the difference between the first formula and this approximation is known as discretization error.
Related phenomena
In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, the analog of discretization is
sampling, and results in no loss if the conditions of the
sampling theorem are satisfied, otherwise the resulting error is called
aliasing.
Discretization error, which arises from finite resolution in the ''domain,'' should not be confused with
quantization error, which is finite resolution in the ''range'' (values), nor in
round-off error arising from
floating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
. Discretization error would occur even if it were possible to represent the values exactly and use exact arithmetic – it is the error from representing a function by its values at a discrete set of points, not an error in these values.
References
See also
*
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
*
Linear multistep method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The proce ...
*
Quantization error
{{DEFAULTSORT:Discretization Error
Numerical analysis