14a the neglected remainder term in 213 n j n

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Unformatted text preview: )n+ uxx j x 0 < < 1 (2.1.4b) is called the local discretization error or local truncation error. Backward nite di erence approximations can be developed by expanding n;1 in a j u 2.1. Di erence Operators 3 Taylor's series about ( ) to obtain 2 n = n ; ( )n + 1 ( 2 )n 2 + j ;1 j j j 2! k +1 k +1 + (;1) 1)! ( k+1 )n; j (+ xj tn u u @u @u x @x x @x @ k + (;1) ( ! k ::: u x @x k k +1 k ) @un @x 0 x kj < < k 1 : (2.1.5) Retaining the rst two terms in (2.1.5) and neglecting the remainder gives the rst backward nite di erence approximation of ( x)n as j u n ( )= Uj n Ux j ; n Uj x ;1 : (2.1.6a) The local discretization error is again obtained from the remainder term as 1 ( )n n 0 1 (2.1.6b) j= 2 xx j; The approximations of ( x)n given by (2.1.4) and (2.1.6) are directional. A centered j nite di erence approximation of the rst derivative can be obtained by retaining the rst three terms ( = 2) in (2.1.2) and (2.1.5) and subtracting the resulting expressions to get u x < < : u k ( )= +1 ; Uj ;1 : n n (2.1.7a) 2 The local discretiz...
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