“Compact stencil”的意思、由来-开放百科全书

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Compact stencil

释义

Two Point Stencil Example
Three Point Stencil Example
See also
References

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:

This is obtained from the Taylor series 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 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 :

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.

1 : Numerical differential equations

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Two Point Stencil Example

Three Point Stencil Example

See also

References