In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the areas of
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 ...
called
numerical partial differential equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In principle, specialized methods for hyperbolic, parabolic or elliptic partia ...
, a compact stencil is a type of
stencil
Stencilling produces an image or pattern on a surface, by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object, to create a pattern or image on a surface, by allowing the pigment to reach ...
that uses only nine nodes for its
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 numerical ...
method in two dimensions. It uses only the center node and the
adjacent
Adjacent or adjacency may refer to:
*Adjacent (graph theory), two vertices that are the endpoints of an edge in a graph
*Adjacent (music), a conjunct step to a note which is next in the scale
See also
*Adjacent angles, two angles that share a c ...
nodes. For any
structured grid
A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks).
Its opposite is irregular grid.
Grids of this type appear on graph paper and may be used in finite element analysis, finite volume ...
utilizing a compact stencil in 1, 2, or 3
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s the maximum number of
nodes
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
is 3, 9, or 27 respectively. Compact stencils may be compared to
non-compact stencils. Compact stencils are currently implemented in many
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's.
[Communications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd.]
Two Point Stencil Example
The two point stencil for the ''first derivative'' of a function is given by:
.
This is obtained from the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the first derivative of the function given by:
.
Replacing
with
, we have:
.
Addition of the above two equations together results in the cancellation of the terms in odd powers of
:
.
.
.
Three Point Stencil Example
For example, the three point stencil for the ''second derivative'' of a function is given by:
.
This is obtained from the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the first derivative of the function given by:
.
Replacing
with
, we have:
.
Subtraction of the above two equations results in the cancellation of the terms in even powers of
:
.
.
.
See also
*
Stencil (numerical analysis)
In mathematics, especially the areas of numerical analysis concentrating on the numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the point of interest by using a numerical a ...
*
Non-compact stencil
In numerical mathematics, a non-compact stencil is a type of discretization method, where any node surrounding the node of interest may be used in the calculation. Its computational time grows with an increase of layers of nodes used. Non-compact ...
*
Five-point stencil
In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations to ...
References
{{reflist
Numerical differential equations