Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms. Part 2 - second-order equations

In this paper, unconditionally stable higher-order accurate time step integration algorithms suitable for linear second-order di!erential equations based on the weighted residual method are presented. The second-order equations are manipulated directly. As in Part 1 of this paper, instead of specifying the weighting functions, the weighting parameters are used to control the algorithm characteristics. The algorithms are at least nth-order accurate if the numerical solution for displacement is approximated by a polynomial of degree n#1 with n undetermined coe$cients. By choosing the weighting parameters carefully, the order of accuracy can be improved. The generalized PadeH approximations for the second-order equations are considered. The ultimate spectral radius k is an algorithmic parameter. By relating the approximate solutions to the equivalent formulations presented in Part 1 of this paper, the required weighting parameters are found explicitly. Any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters. The stabilizing weighting functions for the weighted residual method are found explicitly. To ensure higher-order accuracy in the general solution, the accuracy of the particular solution due to excitation is also examined