Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

Abstract

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.