Time step integration algorithms with predetermined coefficients for second order equations

Abstract

In this paper, the effect of using the predetermined coefficients in constructing time step integration algorithms suitable for linear second order differential equations based on the weighted residual method is investigated. The second order equations are manipulated directly. The displacement approximation is assumed to be in a form of polynomial in the time domain and some of the coefficients can be predetermined from the known initial conditions. The algorithms are constructed so that the approximate solutions are equivalent to the solutions given by the transformed first order equations. If there are m predetermined coefficients (in addition to the two initial conditions) and r unknown coefficients in the displacement approximation, 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. This can be related to the Padé approximations for the second order equations. Unconditionally stable algorithms equivalent to the generalized Padé approximations for the second order equations are presented. The order of accuracy is 2 r−1 or 2 r and it is required that m+1⩽ r. The corresponding weighting parameters, weighting functions and additional weighting parameters for the Padé and generalized Padé approximations are given explicitly.