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Asymptotic rates of convergence of iterative methods applied to finite difference calculations on non‐uniform grids
Author(s) -
Jones I. P.
Publication year - 1983
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620191007
Subject(s) - grid , rate of convergence , convergence (economics) , finite difference , mathematics , finite difference method , iterative method , compression (physics) , simple (philosophy) , diffusion , variable (mathematics) , diffusion equation , mathematical analysis , mathematical optimization , geometry , computer science , physics , thermodynamics , computer network , channel (broadcasting) , philosophy , economy , epistemology , service (business) , economics , economic growth
In this paper the use of strongly stretched finite difference grids is examined in detail for a simple model equation. Numerical solutions of this equation demonstrate the overall second‐order accuracy of the difference approximations. The behaviour of two simple iterative methods, S.O.R. and stationary A.D.I. is discussed for several cases with strong grid stretching and with variable diffusion coefficients. The results show that grid compression near to boundaries can greatly enhance the rates of convergence of these iterative methods, whereas with grid compression in the centre the rate of convergence can be extremely slow. The case of a diffusion coefficient which increases away from the boundaries of a region is analogous to grid compression in the centre and again the convergence rates can degrade considerably.