113b 3 these identities can be used to derive high

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Unformatted text preview: ) approximation of (ux )j . This formula can be veri ed as an ( The formal manipulations used in Examples 2.1.1 - 2.1.3 have to be veri ed as being rigorous. Estimates of local discretization errors must also be obtained. Nevertheless, using the formal operators of Table 2.1.1 provides a simple way of developing high-order nite di erence approximations. O Problems 1. High-order centered di erence approximations can be constructed by manipulating identities involving the central di erence operator ( 3], Chapter 1) uj uj +1=2 ; uj ;1=2 : 1.1. Use Taylor's series expansions of a function ( ) on a uniform mesh of spacing to show that ux x uj where infer Duj 0 (j u +1=2 = e x 2D u j uj ; xD ;1=2 = e 2 uj ). Use the de nition of the central di erence operator to x = 2 sinh 2 x or, inverting, D 2 = 2 sinh;1 = ; 21 3 + 3 5 ; 2 2 3! 245! This relationship can be used to construct central di erence approximations of x. xD ::: : u 1.2. Use the result of Question 1.1 to show that 1 1 ) = 2 ; 12 4 + 90 6 + ] j Truncating this relationship gives higher-order centered di erence approximations of the second...
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