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110b by n uj n uj 1 1 2 2 xl 4 n

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Unformatted text preview: .1. Di erence Operators 5 where max( L R ). Let us also divide (2.1.10a) by L , and add the results to get x x x xR , divide (2.1.10b) by x ( )= n uxx j 2 R+ x +1 ; uj n uj xL n xR n ; uj ; n uj xL ;1 ] ; 1( 6 xR ; xL )( )n + ( 2) (2.1.11b) uxxx j O x : The approximations of ( x)n and ( xx)n that are obtained by retaining the rst terms j j of (2.1.11a) and (2.1.11b) are only accurate to ( ). If L = R , the xx terms in (2.1.11a) and the xxx terms in (2.1.11b) cancel and the accuracy of both formulas is ( 2). It will be convenient to have a shorthand operator notation for the nite di erence operators in the same way that such notation is used for derivatives. The notation shown in Table 2.1.1 is relatively standard and will be used throughout these notes. For simplicity, we have suppressed the time dependence and only show spatial di erence operators in Table 2.1.1. Thus, we have assumed that is a function of only with j ( ). Temporal di erence operators are de ned in analogous fashion. Some examples follow. u u O x x x u u O x u...
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