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. 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. The two point stencil for the first derivative of a function is given by: f ′ ( x 0 ) = f ( x 0 + h ) − f ( x 0 − h ) 2 h + O ( h 2 ) {displaystyle f'(x_{0})={frac {fleft(x_{0}+h ight)-fleft(x_{0}-h ight)}{2h}}+Oleft(h^{2} ight)} . This is obtained from the Taylor series expansion of the first derivative of the function given by: f ′ ( x 0 ) = f ( x 0 + h ) − f ( x 0 ) h − f ( 2 ) ( x 0 ) 2 ! h − f ( 3 ) ( x 0 ) 3 ! h 2 − f ( 4 ) ( x 0 ) 4 ! h 3 + ⋯ {displaystyle {egin{array}{l}f'(x_{0})={frac {fleft(x_{0}+h ight)-f(x_{0})}{h}}-{frac {f^{(2)}(x_{0})}{2!}}h-{frac {f^{(3)}(x_{0})}{3!}}h^{2}-{frac {f^{(4)}(x_{0})}{4!}}h^{3}+cdots end{array}}} . Replacing h {displaystyle h} with − h {displaystyle -h} , we have: f ′ ( x 0 ) = − f ( x 0 − h ) − f ( x 0 ) h + f ( 2 ) ( x 0 ) 2 ! h − f ( 3 ) ( x 0 ) 3 ! h 2 + f ( 4 ) ( x 0 ) 4 ! h 3 + ⋯ {displaystyle {egin{array}{l}f'(x_{0})=-{frac {fleft(x_{0}-h ight)-f(x_{0})}{h}}+{frac {f^{(2)}(x_{0})}{2!}}h-{frac {f^{(3)}(x_{0})}{3!}}h^{2}+{frac {f^{(4)}(x_{0})}{4!}}h^{3}+cdots end{array}}} . Addition of the above two equations together results in the cancellation of the terms in odd powers of h {displaystyle h} : 2 f ′ ( x 0 ) = f ( x 0 + h ) − f ( x 0 ) h − f ( x 0 − h ) − f ( x 0 ) h − 2 f ( 3 ) ( x 0 ) 3 ! h 2 + ⋯ {displaystyle {egin{array}{l}2f'(x_{0})={frac {fleft(x_{0}+h ight)-f(x_{0})}{h}}-{frac {fleft(x_{0}-h ight)-f(x_{0})}{h}}-2{frac {f^{(3)}(x_{0})}{3!}}h^{2}+cdots end{array}}} .