Compact stencil
Encyclopedia
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the areas of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
called numerical partial differential equations
Numerical partial differential equations
Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations .Numerical techniques for solving PDEs include the following:...
, a compact stencil is a type of stencil
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 approximation routine. Stencils are the basis for...
that uses only nine nodes for its discretization
Discretization
In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
method in two dimensions. It uses only the center node and the adjacent
Adjacent
Adjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
nodes. For any structured grid utilizing a compact stencil in 1, 2, or 3 dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s the maximum number of nodes 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
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's.
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
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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
:
.
.
.