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Unformatted text preview: trary function ( ) using a linear
combination of the “weighting” functions. Then one chooses the expansion coefficients so that ∫ (() ∑ ( ⃗ )) , then ( ) must be “small”. is minimized. If this procedure yields Let us now derive the standard finite volume method for the diffusion equation. Our first task is to
derive some difference formulas using Taylor series expansions.
( ) () ( ) () From the top equation, we get () () () () () () ( ) ( ) ( ) () () () () () () () ( ) ( ) From the bottom equation, we get
() ( ) The first difference formula is called a forward difference. Not that it becomes exact as
leading order error proportional to . with the The second difference formula is called a backward difference. Note that it has accuracy properties
similar to those of the forward difference operator.
If we average the forward and backward difference formulae, we get the central difference formula:
( ) ( ) () () ( ) Note that this formula is more accurate than the other two, becau...
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This note was uploaded on 10/06/2013 for the course NEUN 430 taught by Professor Morel during the Fall '10 term at Texas A&M.
 Fall '10
 Morel

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