Taylor_Tables

Taylor_Tables - Taylor Tables of Di erencing Schemes 1....

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Unformatted text preview: Taylor Tables of Di erencing Schemes 1. Notation: Consider u ( x; t ) for xed t and x = j ¡ x so that, u ( x + k ¡ x ) = u ( j ¡ x + k ¡ x ) = u j + k . 2. The generalized form of the Taylor Series Expansions is given by u j + k = u j + ( k ¡ x ) @u @x j + 1 2 ( k ¡ x ) 2 @ 2 u @x 2 j + : : : + 1 n ! ( k ¡ x ) n @ n u @x n j + : : : 3. For example, consider the Taylor series expansion for u j +1 : u j +1 = u j + (¡ x ) @u @x j + 1 2 (¡ x ) 2 @ 2 u @x 2 j + : : : + 1 n ! (¡ x ) n @ n u @x n j + : : : 4. Or for u j 2 : u j 2 = u j + ( 2¡ x ) @u @x j + 1 2 ( 2¡ x ) 2 @ 2 u @x 2 j + : : : + 1 n ! ( 2¡ x ) n @ n u @x n j + : : : 1 Taylor Table For the 1 st Order Backward Di erence 1. Given @u @x j ( u j u j 1 ) ¡ x = er t 2. Each term expanded in Taylor Series and placed in a table simpli ng algebra. 3. Note the multiplication by ¡ x to again simplify the table....
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This note was uploaded on 04/30/2010 for the course MAE 107 taught by Professor Rottman during the Fall '08 term at UCSD.

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Taylor_Tables - Taylor Tables of Di erencing Schemes 1....

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