Comparison of Various Orthogonal Polynomials in hp-Version Time Finite Element Method

Four types of orthogonal polynomials-Legendre, Chebyshev, Hermite, and integrated Legendre-are evaluated as basis functions in the time finite element method to solve initial value problems governed by second-order differential equations. Condition numbers of the augmented stiffness matrix for the selected problems are estimated by increasing the number of polynomial terms in the expansion. Results for the CPU time using an IBM 3090-300E/VF computer and the estimated condition numbers, using IMSL subroutine DLFCRG, for each of four basis functions are presented. The augmented stiffness matrix for the case of integrated Legendre polynomials is found to have the best condition number behavior, and that for the Hermite polynomial the worst. The integrated Legendre polynomials require the most CPU time, whereas the Chebyshev polynomials require the least.