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Unformatted text preview: tal predictor sweep requires the solution of K ; 1 tridiagonal systems of
dimension J ; 1. Each tridiagonal system requires approximately 5J operations, where
an operation is one multiplication or division plus one addition or subtaction (cf. Section
4.1). Thus, the predictor step requires approximately 5JK operations. Similarly, in the
corrector step, we have to solve J ; 1 tridiagonal systems of dimension K ; 1, which also
require approximately 5JK operations. Therefore, the total operation count per time
step is 10JK operations, which would normally be far less than the (5=3)KJ 3 operations
needed for the block tridiagonal algorithm.
n
j n y xj = 8 MultiDimensional Parabolic Problemss The local discretization error for multidimensional problems is de ned exactly the
same as for onedimensional problems (De nition 3.1.1). The intermediate ADI solution
introduces an added complication thus, we have to either combine separate estimates
of the local discretization errors of the predictor and corrector steps or eliminate U +1 2
from (5.1.5b). The latter course is the simpler of the two for this application since U +1 2
may be eliminated by adding and subtracting (5.1.5a) and (5.1.5b). The result is
n = jk
n = jk U n +1 ; U jk n
jk = r 2 U +1 2 + r 2(U +1 + U )
2
n x x y = n n jk y jk jk = 1 (U +1 + U ) ; r 2 (U +1 ; U ):
2
4
Substituting the second equation into the rst
1
U +1 ; U = 2 (r 2 + r 2)(U +1 + U ) ; r 4r 2 2(U +1 ; U ):
Dividing by t, gathering all terms on the right side, replacing the numerical approximation by any smooth function, e.g., the exact solution of the di erential equation, and
subtracting the result from the di erential equation (5.1.2a) yields the local discretization
error as U n = n jk y n jk jk n jk +1 2 n jk n x y x n jk x n jk y n jk y y jk n x y n jk jk (1 ; r 2 ; r 2 )u +1 + (1 + r 2 + r 2)u
2
2
2
2
r r 2 2(u +1 ; u ):
;
4
Remark 1. The term t is not the local error. Since this scheme is implicit, the
expression for the local error is more complex.
The rst three terms of the above expression are the product of t and the local
discretization error of the CrankNicolson scheme (5.1.4a), i.e., t n = t(u u t ; u) xx ; n yy jj k x ; x jk y y x n x y x jk n y jk n jk y x n y jk n jk (1 ; r2 2 ; r2 2)u +1 + (1 + r2 2 + r2 2 )u :
A Taylor's series expansion would reveal that t( )
n jk CN = t(u t ; u ; xx () u) n jk CN n jj yy k x ; y x x n y y x jk = O( x2) + O( y2) + O( t2 ): Expanding the remaining term in a Taylor's series yields rr
x 4 y 2 2 (u +1 ; u
n x y n jk jk 2
) = r 4r 2 2 u +1 2 = 4 t...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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