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Unformatted text preview: u i, j ,i.e. 1 u i1, j u i1, j 2 (4.3) Substituting relation (4.3) in equation (4.2) ,we get u i, j1 1 2r
2r
u i. j1 u i1, j u i1, j (4.4) 1 2r
1 2r This scheme is called Dufort Frankel Explicit scheme.
The advantage of this scheme is that inspite of it being an Explicit scheme, it has truncation
error o( t) 2 o(x) 2 . But on the other hand, it requires the solution at the first time level to be determined by any other two time level scheme. For j = 2 onwards, this scheme can be
applied.
The computational molecule of this scheme is as shown in fig(2). u i,j+1 (j+1)th level jth level 2 r 2r u i, j u i+1,j u i‐1,j (j‐1)th level u i,j‐1 Fig (2) These schemes however have some problem with stability and compatibility which will be
discussed later. Example: Solve the partial differential equation : u 2u t dx 2
Subject to ; 0 x 1, t 0 u x,0 x,u 0,t 0 ; u 1, t 1,
With r 0.75, x 0.1 Obtain the solution at first time level using CN scheme and obtain the second time level by
(i) Dufort Frankel (ii) Richardson
and compare the result. Solution: By Crank Nicolson, equation (1) can be approximated as ru i1, j1 (2 2r...
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 Spring '14
 Prof.RamaBhargava
 Differential Equations, Equations, Derivative, Partial Differential Equations

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