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Two practical approaches to estimating the local and

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Unformatted text preview: Cn 1=2 Cn), the local errors of both solutions are ; ; ; ; ; ; ; ; ; h yn ; y(tn) = Cnhp+1 + O(hp+2) and h= yn 2 ; y(tn) = 2Cn(h=2)p+1 + O(hp+2) Subtracting the two solutions to eliminate the exact solution gives h h yn ; yn =2 = Cnhp+1(1 ; 2 p) + O(hp+2): ; Neglecting the O(hp+2) term, we estimate the local error in the solution of (3.5.2a) as h=2 h n;y jdnj jCnjhp+1 = jy1 ; 2 np j : ; (3.5.3a) Computation of the error estimate requires 2s additional function evaluations (to compute yn 1=2 and yn) for an s-stage Runge-Kutta method. If s p then approximately ; 44 2p extra function evaluations (for scalar systems). This cost for m-dimensional vector problems is approximately 2pm function evaluations per step. Richardson's extrapolation is particularly expensive when used with implicit methods because the change of step size requires another Jacobian evaluation and (possible) factorization. It may, however, be useful with DIRK methods because of their lower triangular coe cient matrices. h= It's possible to estimate the error of the solution yn 2 as h= p+1 jy h ; yn 2j n h=2 j jCn jh jdn (3.5.3b) 2p = 2p ; 1 : h Proceeding in this manner seems better than accepting yn as the solution however, it is a bit risky since we do not have an estimate of the error of the intermediate solution h= yn 21=2. h= h Finally, th...
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