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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
u 2.1. Di erence Operators 3 Taylor's series about ( ) to obtain
= n ; ( )n + 1 ( 2 )n 2 +
+ (;1) 1)! ( k+1 )n;
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
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
2 xx j;
The approximations of ( x)n given by (2.1.4) and (2.1.6) are directional. A centered
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
u x < < : u k ( )= +1 ; Uj ;1 : n n (2.1.7a)
The local discretiz...
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- Spring '14