By Franz L. Alt (Ed.)

ISBN-10: 0120121026

ISBN-13: 9780120121021

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29) [)(xi, t n + i , i ( w i , n + l win)) + + + + + + NUMERICAL METHODS FOR PARABOLIC EQUATIONS 23 'I'hc soliitioii of the slgchraic proldcm may hr ol,tninctl hy itcmtiiig on w,, I 1, using thr old itrratc in tlic r1v:iliint ion of t hc corfficienth. If an arciirule cstimatc of thc solutioii at timc l n + : could ho provided, it, \vould 1)c unnecessary to iteratc a t all. The author [30] proposcd to usc the forward difference equation to predict wn++ from w,; however, the proof offered was incornpletc.

Thus, as Ax and At tend to zero, the error tends to zero in the grid LZnorm. Of course, this is not the usual integral L2topology; however, it can be shown [27] using interpolation to define the solution over the rectangle (0 5 x 5 1,0 5 t 5 T) that the integral Lz norm of the error is also O ( ( A X ) ~ ( A t ) z ) . Note that we have preserved the local accuracy in the global error. The above argument is a special case of this method of the author [27]. It will be shown later that the optimal relation between A,?

3) Now, if r is large, the magnitude of t,he stability ratio can be made large by t'aking q = M - 1 and p = 1 ; hence, unlimited stability does not result from (10,l). 4) would lead to a stability ratio in which the positions of p and q would be JIM DOUGLAS, JR. 38 i n t erkhaiigcd. 4). As the cigcnfunrtioiis fire 11ie same for both rclnfions, flw stability ratioii for tlic doiiblc stcp is pt1+2 = 1 - 4r sin? ( ~ p A x / 2 ) 1 - 4 sill” ( n y A z / 2 (I 4r sin? ( ~ p A x / 2 ) 1 4 sin2 ( r q A z / $ + + which is bounded in magnitude by one for any size time step.

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Advances in Computers, Vol. 2 by Franz L. Alt (Ed.)


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