Weighting parameters for time-step integration algorithms with predetermined coefficients

In this paper, the e ect of using the predetermined coe cients in constructing time-step integration algo- rithms is investigated. Both rst- 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 Taylorʹs series expansion of the exact solution. Therefore, some of the coe cients can be predetermined from the known initial conditions. If there are m predetermined coe cients and r unknown coe cients 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 m6r for rst-order equations and m + 16r for second-order equations. Hence, the Dahlquistʹs theorem is generalized. Algorithms equivalent to the Pad e approximations and uncondition- ally stable algorithms equivalent to the generalized Pad e approximations are constructed. The corresponding weighting parameters and weighting functions for the Pad e and generalized Pad e approximations are given explicitly