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Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics
Author(s) -
Cheer A.,
Saleem M.
Publication year - 1991
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650120503
Subject(s) - mathematics , preconditioner , eigenvalues and eigenvectors , rate of convergence , power iteration , convergence (economics) , acceleration , generalized minimal residual method , computational fluid dynamics , iterative method , mathematical analysis , algorithm , computer science , physics , computer network , channel (broadcasting) , quantum mechanics , classical mechanics , mechanics , economics , economic growth
Eigensystem analysis techniques are applied to finite difference formulations of the Navier‐Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix. On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. This procedure has improved the rates of convergence of CFD codes (developed at NASA Ames Research Center) by 20%–50%. The sensitivity of the computed solution with respect to the shift is also studied. Finally, an adaptive shifting of the spectrum together with Wynn's acceleration algorithm are discussed. It turns out that the shifting process is a preconditioner for Wynn's method.