For example a centered di erence approximation of xxn

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Unformatted text preview: ons to get u n ; 2 jn + jn;1 )n = j+1 2 ( U U Uxx j (2.1.9a) U : x The discretization error of this approximation is n j 1 = ; 12 ( )n+ uxxxx j x ;1 2 < < 1 (2.1.9b) where the of (2.1.9) is a generic symbol and has no relation to the used in (2.1.3) or (2.1.4b). Centered di erences only have a higher order of accuracy than forward or backward di erences on uniform grids. To see this, consider three points j;1, j , and j+1 of a nonuniform grid as shown in Figure 2.1.2. Let L j ; j ;1 and R j +1 ; j and construct the Taylor's series expansions x x x x x x x x x n+1 ∆x L ∆x R n j-1 j j+1 Figure 2.1.2: Two neighboring intervals of a nonuniform grid. n + ;1 = uj ; n uj Divide (2.1.10a) by n xR n ; ( x)n = 1 j+1 j 2 R u u ( )n + 1 2 +1 = uj n uj x xR ux j ( )n + 1 2 xL ux j , divide (2.1.10b) by n uj + n uj ; ;1 ] ; 1 ( 2( 2( xL 4 )n + ( 3 ) L O xL uxx j xL n uj )n + ( 3 ) R xR uxx j xR O (2.1.10a) x x (2.1.10b) : , and subtract the results to get ; xL )( )n + ( 2 ) uxx j O x : (2.1.11a) 2...
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