Compact stencil
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] The two point stencil for the first derivative of a function is given by:
This is obtained from the Taylor series expansion of the first derivative of the function given by:
{\displaystyle {\begin{array}{l}f'(x_{0})={\frac {f\left(x_{0}+h\right)-f(x_{0})}{h}}-{\frac {f^{(2)}(x_{0})}{2!
}}h^{3}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f'(x_{0})=-{\frac {f\left(x_{0}-h\right)-f(x_{0})}{h}}+{\frac {f^{(2)}(x_{0})}{2!
}}h^{3}+\cdots \end{array}}}
Addition of the above two equations together results in the cancellation of the terms in odd powers of
{\displaystyle {\begin{array}{l}2f'(x_{0})={\frac {f\left(x_{0}+h\right)-f(x_{0})}{h}}-{\frac {f\left(x_{0}-h\right)-f(x_{0})}{h}}-2{\frac {f^{(3)}(x_{0})}{3!
}}h^{2}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f'(x_{0})={\frac {f\left(x_{0}+h\right)-f\left(x_{0}-h\right)}{2h}}-{\frac {f^{(3)}(x_{0})}{3!
}}h^{2}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f'(x_{0})={\frac {f\left(x_{0}+h\right)-f\left(x_{0}-h\right)}{2h}}+O\left(h^{2}\right)\end{array}}}
For example, the three point stencil for the second derivative of a function is given by:
{\displaystyle {\begin{array}{l}f^{(2)}(x_{0})={\frac {f\left(x_{0}+h\right)+f\left(x_{0}-h\right)-2f(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:
{\displaystyle {\begin{array}{l}f'(x_{0})={\frac {f\left(x_{0}+h\right)-f(x_{0})}{h}}-{\frac {f^{(2)}(x_{0})}{2!
}}h^{3}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f'(x_{0})=-{\frac {f\left(x_{0}-h\right)-f(x_{0})}{h}}+{\frac {f^{(2)}(x_{0})}{2!
}}h^{3}+\cdots \end{array}}}
Subtraction of the above two equations results in the cancellation of the terms in even powers of
{\displaystyle {\begin{array}{l}0={\frac {f\left(x_{0}+h\right)-f(x_{0})}{h}}+{\frac {f\left(x_{0}-h\right)-f(x_{0})}{h}}-2{\frac {f^{(2)}(x_{0})}{2!
}}h^{3}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f^{(2)}(x_{0})={\frac {f\left(x_{0}+h\right)+f\left(x_{0}-h\right)-2f(x_{0})}{h^{2}}}-2{\frac {f^{(4)}(x_{0})}{4!
}}h^{2}+\cdots \end{array}}}
{\displaystyle {\begin{array}{l}f^{(2)}(x_{0})={\frac {f\left(x_{0}+h\right)+f\left(x_{0}-h\right)-2f(x_{0})}{h^{2}}}+O\left(h^{2}\right)\end{array}}}
