15b for all interior points in column j gives n k i

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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 Multi-Dimensional Parabolic Problemss The local discretization error for multi-dimensional problems is de ned exactly the same as for one-dimensional 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 Crank-Nicolson 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.

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