12 and 215 and subtracting the resulting expressions

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Unformatted text preview: ation error is obtained from the remainder terms as 1 n 2 n ;1 1 (2.1.7b) j = ; ( xxx )j + 6 The discretization error of the centered formula (2.1.7) is ( 2) while those of the forward and backward formulas (2.1.4) and (2.1.6) are ( ). Since the centered formula converge at a faster rate than either of the two directional formulas, it would normally be preferred however, we shall see examples (Section 2.2) where this is not the case. Obviously, Taylor's series can also be used to construct approximations of time derivatives. The rst forward di erence approximation of t at ( j n) is n Ux j Uj u x x < < : O O u ()= n Ut j +1 ; U n n Uj j t x x x t (2.1.8a) : The local discretization error is n j = ; 1 ( tt )n+ 2j u t 0 < < 1 : (2.1.8b) 4 Finite Di erence Methods The same approach can be used to construct approximations of higher derivatives. For example, a centered di erence approximation of ( xx)n can be obtained by retaining j the rst four terms in (2.1.2) and (2.1.5) and adding the resulting expressi...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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