Weighting parameters for time-step integration algorithms with predetermined coefficients

Abstract

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+2r. 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 Pade approximations and unconditionally stable algorithms equivalent to the generalized Pade approximations are constructed. The corresponding weighting parameters and weighting functions for the Pade and generalized Pade approximations are given explicitly. Copyright © 2000 John Wiley & Sons, Ltd.