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Partial Differential Equations part 6

The best way to solve equations of the form (19.4.28), including the constant coefficient problem (19.0.3), is a combination of Fourier analysis and cyclic reduction, the FACR method [3-6]. If at the rth stage of CR we Fourier analyze the equations of the form (19.4.32) along y | 19.5 Relaxation Methods for Boundary Value Problems 863 FACR Method The best way to solve equations of the form 19.4.28 including the constant coefficient problem 19.0.3 is a combination of Fourier analysis and cyclic reduction the FACR method 3-6 . If at the rth stage of CR we Fourier analyze the equations of the form 19.4.32 along y that is with respect to the suppressed vector index we will have a tridiagonal system in the . --direction for each y-Fourier mode p k I A rKtk A2n r k 19 4 35 Uj 2r I A Uj Uj 2r A gj 19.4.35 Here xj is the eigenvalue of T r corresponding to the kth Fourier mode. For the equation 19.0.3 equation 19.4.5 shows that A j will involve terms like cos 2 k L - 2 raised to a power. Solve the tridiagonal systems for uj at the levels j 2r 2 x 2r 4 x 2r . J - 2r. Fourier synthesize to get the y-values on these -lines. Then fill in the intermediate -lines as in the original CR algorithm. The trick is to choose the number of levels of CR so as to minimize the total number of arithmetic operations. One can show that for a typical case of a 128 x 128 mesh the optimal level is r 2 asymptotically r log2 log2 J . A rough estimate of running times for these algorithms for equation 19.0.3 is as follows The FFT method in both and y and the CR method are roughly comparable. FACR with r 0 that is FFT in one dimension and solve the tridiagonal equations by the usual algorithm in the other dimension gives about a factor of two gain in speed. The optimal FACR with r 2 gives another factor of two gain in speed. CITED REFERENCES AND FURTHER READING Swartzrauber P.N. 1977 SIAM Review vol. 19 pp. 490-501. 1 Buzbee B.L Golub G.H. and Nielson C.W. 1970 SIAMJournal on NumericalAnalysis vol. 7 pp. 627-656 see also op. cit. vol. 11 pp. 753-763. 2 Hockney R.W. 1965 Journal ofthe Association for Computing Machinery vol. 12 pp. 95-113. 3 Hockney R.W. 1970 in Methods of Computational Physics vol. 9 New York Academic Press pp. 135-211. 4 Hockney R.W. and Eastwood J.W. 1981 .