# 113b 3 these identities can be used to derive high

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online