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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
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
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
u u O x x x u u O x u...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14