Abstract
This paper reviews recent works on acceleration techniques for iterative solutions of elliptic and mixed-type problems, using algorithms related to Padé's fractions. The study focuses on the question of how to speed up convergence of relaxation methods currently available for transonic and related flow computations, with minimal alterations in computer programming and storage requirements.
The theoretical basis of the work is similar to the power method, but allowance is made that moduli of some of the eigen-values can be very close to one another and to unity. The study contributes to a clarification of the error analyses for the sequence transformations of Aitken, Shanks,and Wilkinson, and to developing a cyclic iterative procedure applying the transformations to accelerating large linear and nonlinear systems. Use of the first and second order transforms similar to Shanks' (corresponding to the second and third rows in the upper half of Padé's Table) is shown to be effective, but their subtle differences from the latter prove to be crucial.
Examples illustrating the accelerating technique include transonic flow as well as model Dirichlet problems. Reduction by a factor of three to five in computing time is possible, depending on the accuracy requirement and the order of the transformation. The possibility for reducing the computer storage requirement via Wynn's recursive identities is examined for a linear system in Appendix A.
This research was supported by the Office of Naval Research under Contract Number N00014-67-A-0269-0021.
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Cheng, H.K., Hafez, M.M. (1976). Cyclic iterative method applied to transonic flow analyses. In: Cabannes, H. (eds) Padé Approximants Method and Its Applications to Mechanics. Lecture Notes in Physics, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015662
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DOI: https://doi.org/10.1007/BFb0015662
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