Compact stencil explained

In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compact stencil in 1, 2, or 3 dimensions 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 solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's.^{[1] [2]}

Two Point Stencil Example

The two point stencil for the first derivative of a function is given by:

f'(x _{0 ) = (f\left(x _{0} + h\right) - f\left(x _{0} - h\right))/(2 h)} + O\left(h ^{2\right)} .

This is obtained from the Taylor series expansion of the first derivative of the function given by:

\begin{array} {l} f'(x _{0 ) = (f\left(x _{0} + h\right) - f(x _{0)})/(h)} - (f ^{(2)} (x _{0)})/(2 !) h - (f ^{(3)} (x _{0)})/(3 !) h ^{2} - (f ^{(4)} (x _{0)})/(4 !) h ^{3} + … \end{array} .

Replacing

h with - h , we have:\begin{array} {l} f'(x _{0 ) =- (f\left(x _{0} - h\right) - f(x _{0)})/(h)} + (f ^{(2)} (x _{0)})/(2 !) h - (f ^{(3)} (x _{0)})/(3 !) h ^{2} + (f ^{(4)} (x _{0)})/(4 !) h ^{3} + … \end{array} .

Addition of the above two equations together results in the cancellation of the terms in odd powers of

h :\begin{array} {l} 2 f'(x _{0 ) = (f\left(x _{0} + h\right) - f(x _{0)})/(h)} - (f\left(x _{0} - h\right) - f(x _{0)})/(-2 h) (f ^{(3)} (x _{0)})/(3 !) h ^{2} + … \end{array} .\begin{array} {l} f'(x _{0 ) = (f\left(x _{0} + h\right) - f\left(x _{0} - h\right))/(2 h)} - (f ^{(3)} (x _{0)})/(3 !) h ^{2} + … \end{array} .\begin{array} {l} f'(x _{0 ) = (f\left(x _{0} + h\right) - f\left(x _{0} - h\right))/(2 h)} + O\left(h ^{2\right) \end{array}} .

Three Point Stencil Example

For example, the three point stencil for the second derivative of a function is given by:

\begin{array} {l} f ^{(2)} (x _{0 ) = (f\left(x _{0} + h\right) + f\left(x _{0} - h\right) -2 f(x _{0)})/(h ^{2})} + O\left(h ^{2\right) \end{array}} .

This is obtained from the Taylor series expansion of the first derivative of the function given by:

\begin{array} {l} f'(x _{0 ) = (f\left(x _{0} + h\right) - f(x _{0)})/(h)} - (f ^{(2)} (x _{0)})/(2 !) h - (f ^{(3)} (x _{0)})/(3 !) h ^{2} - (f ^{(4)} (x _{0)})/(4 !) h ^{3} + … \end{array} .

Replacing

h with - h , we have:\begin{array} {l} f'(x _{0 ) =- (f\left(x _{0} - h\right) - f(x _{0)})/(h)} + (f ^{(2)} (x _{0)})/(2 !) h - (f ^{(3)} (x _{0)})/(3 !) h ^{2} + (f ^{(4)} (x _{0)})/(4 !) h ^{3} + … \end{array} .

Subtraction of the above two equations results in the cancellation of the terms in even powers of

h :\begin{array} {l} 0= (f\left(x _{0} + h\right) - f(x _{0)})/(+ h) (f\left(x _{0} - h\right) - f(x _{0)})/(h) -2 (f ^{(2)} (x _{0)})/(2 !) h -2 (f ^{(4)} (x _{0)})/(4 !) h ^{3} + … \end{array} .\begin{array} {l} f ^{(2)} (x _{0 ) = (f\left(x _{0} + h\right) + f\left(x _{0} - h\right) -2 f(x _{0)})/(h ^{2})} -2 (f ^{(4)} (x _{0)})/(4 !) h ^{2} + … \end{array} .\begin{array} {l} f ^{(2)} (x _{0 ) = (f\left(x _{0} + h\right) + f\left(x _{0} - h\right) -2 f(x _{0)})/(h ^{2})} + O\left(h ^{2\right) \end{array}} .

See also

• Stencil (numerical analysis)
• Non-compact stencil
• Five-point stencil

Notes and References

1. W. F. Spotz. High-Order Compact Finite Difference Schemes for Computational Mechanics. PhD thesis, University of Texas at Austin, Austin, TX, 1995.
2. Communications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd.