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Weighting parameters for time‐step integration algorithms with predetermined coefficients
Author(s) -
Fung T. C.
Publication year - 2000
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 - 10.1002/1097-0207(20001220)49:11<1397::aid-nme8>3.0.co;2-3
Subject(s) - weighting , mathematics , algorithm , taylor series , series (stratigraphy) , residual , polynomial , method of mean weighted residuals , mathematical analysis , nonlinear system , medicine , paleontology , physics , quantum mechanics , galerkin method , biology , radiology
In this paper, the effect of using the predetermined coefficients in constructing time‐step integration algorithms is investigated. Both first‐ and second‐order equations are considered. The approximate solution is assumed to be in a form of polynomial in the time domain. It can be related to the truncated Taylor's series expansion of the exact solution. Therefore, some of the coefficients can be predetermined from the known initial conditions. If there are m predetermined coefficients and r unknown coefficients in the approximate solution, the unknowns can be solved by the weighted residual method. The weighting parameter method is used to investigate the resultant algorithm characteristics. It is shown that the formulation is consistent with a minimum order of accuracy m + r . The maximum order of accuracy achievable is m +2 r . Unconditionally stable algorithms exist if m ⩽ r for first‐order equations and m +1⩽ r for second‐order equations. Hence, the Dahlquist's theorem is generalized. Algorithms equivalent to the Padé approximations and unconditionally stable algorithms equivalent to the generalized Padé approximations are constructed. The corresponding weighting parameters and weighting functions for the Padé and generalized Padé approximations are given explicitly. Copyright © 2000 John Wiley & Sons, Ltd.