Compact stencil

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A 2D compact stencil using all 8 adjacent nodes, plus the center node (in red).

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[edit]

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

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

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Replacing with , we have:

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

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.

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Three Point Stencil Example[edit]

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

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

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Replacing with , we have:

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

.

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See also[edit]

References[edit]

  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.