UNCONDITIONALLY STABLE COLLOCATION ALGORITHMS FOR SECOND ORDER INITIAL VALUE PROBLEMS

Abstract

In this paper, unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented. The second order equations are manipulated directly. If the approximate solution is expressed as a polynomial of degree n+1, there are n unknowns to be determined after taking into account the two given initial conditions. It is well known that by suppressing the residuals of the governing equations at n distinct collocation points only, the resultant algorithms are only conditionally stable. In this paper, linear combinations of the residuals at n+1 distinct collocation points are used to solve for the n unknowns. The collocation points and the relative weights between the residuals are derived from the weighted residual method. The weighting functions are arbitrary polynomials of degree not exceeding n−1. To control the accuracy and stability properties of the resultant algorithms, the reduced integration technique is used to evaluate the integrals in the formulation. Once the reduced integration rules are decided, the equivalent collocation form can be derived. It is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non-linear problems directly. Numerical examples are given to illustrate the validity of the present formulation.