Di erence approximations may be constructed in a

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Unformatted text preview: mations may be constructed in a variety of ways, but the use of Taylor's formula is probably the simplest for our present purposes. To begin, let us use Taylor's formula to express n+1 in terms of n and its derivatives as j j 2 1 ( )n 2 + + 1 ( k )n k n n n )j + 2! 2 j j +1 = j + ( ! kj k +1 + 1 ( k+1 )n+ k+1 (2.1.2) j ( + 1)! u u x x U t u t u u u @u @x x u @u @x x ::: @ k 1 @x @u k u x @x x : 2 Finite Di erence Methods t n (j,n) 2 1 -2 -1 0 1 2 j x Figure 2.1.1: A partition of the upper half of the ( )-plane into uniform cells of size . xt x t The last term in (2.1.2), the remainder, involves the evaluation of k+1 k+1 at = ( + ) and , with an unknown point on (0 1). As an example, suppose that we retain the rst two terms ( = 1) in (2.1.2) and solve for ( )n to obtain j @u j x n =@x x t k @u=@x ( )= n ux j +1 ; uj n uj n ; 1 ( xx)n+ j 2 u x (2.1.3) x: Neglecting the remainder term, we get the formula for the rst forward nite di erence approximation of x as u +1 ; Uj ( )= n Uj n Ux j n x (2.1.4a) : The neglected remainder term in (2.1.3) n j n ; ( x)n ; j+1 j u u x n uj = ;1( 2...
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