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CFD lecture 17 - Computational Fluid Dynamics Lecture 17...

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Computational Fluid Dynamics Lecture 17 Stopping Criteria Continued There is a solution that is as accurate as possible with a given difference equation and grid. The number of iterations required to improve the solution one decimal place (for the Jacobi scheme) is approximately ( ) 10 1 10 1 log 0.1 where log m λ λ = is the largest eigenvalue of the error propogation matrix. ( ) 1 1 1 1 1 4 if 0.5 22 if 0.9 230 if 0.99 2302 if 0.999 or the iteration matrix. k k k k m e e BAe I BA e R I BA λ λ λ λ + = = = = = = = = K K K K R is the convergence factor 1 1 1 L k k k j j j e a λ + + = = = K K j x K e R . For example, for a 2-D problem with 100, 100 x y N N = = it takes many more iterations than for with for the Jacobi scheme it should take: 10, 10 x y N N = = 100 x y N N = = 1 10 10 46 0.951057 100 100 4666 0.99507 1000 1000 466,601 0.999995 10 100 2,365 0.999027 50 100 2,916 0.999211 50 1000 233,922 0.999990 x y N N λ m asymptotic results Initial convergence is faster than asymptotic result. This is an upper bound. If the error is in the direction of a smaller eigenvalue then convergence will be faster.
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