Unconditionally stable higher-order accurate collocation time-step integration algorithms for first-order equations Original Research Article

In this paper, unconditionally stable higher-order accurate time-step integration algorithms for linear first-order differential equations based on the collocation method are presented. The amplification factor at the end of the spectrum is a controllable algorithmic parameter. The collocation parameters for unconditionally stable higher-order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor. In general, when the numerical solution is approximated by a polynomial of degree n, this approximation is at least nth order accurate. However, by using the above collocation parameters, the order of accuracy can be improved to 2n−1 or 2n. The approximate solutions are found to be equivalent to the generalized Padé approximations. Furthermore, it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions. No modification of the collocation parameters is required.