For simplicity we have suppressed the time dependence

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Unformatted text preview: x u ux Operator Symbol De nition Forward Di erence j j +1 ; j Backward Di erence r r j j ; j;1 Central Di erence j j +1=2 ; j ;1=2 Average ( j+1=2 + j;1=2) 2 j Shift j j +1 Derivative ( x)j j Table 2.1.1: De nition of nite di erence operators. u u u u u E u u u u u Eu D u Du u = u u Example 2.1.1. The centered di erence formula (2.1.7a) can be expressed in terms of the central di erence and averaging operators and . Observe that uj = ( j+1=2 ; j;1=2) = 1 ( j+1 ; j;1) 2 u u Thus, uj x u = j+1 ; j;1 2 u u x : u : 6 Finite Di erence Methods Example 2.1.2. An operator appearing to a positive integral power is iterated thus, 2u j = ( j+1=2 ; j;1=2) = j+1 ; 2 j + j;1 u u u u u : Thus, the centered second di erence approximation (2.1.9a) of the second derivative can be written as 2j ( xx)j 2 u u x : Example 2.1.3. Let us expand uj +1 in a Taylor's series about xj to obtain 1 x2 (u ) + : : : uj +1 = uj + x(ux )j + xx j 2 or, using the derivative operator D de ned in Table 2.1.1, 1 x2 D2 + : : : ]u : uj +1 = 1 + xD + j 2 We may use the Taylor's series expansion...
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