Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms. Part 1 - first-order equations

In this paper, unconditionally stable higher-order accurate time step integration algorithms suitable for linear "rst-order di!erential equations based on the weighted residual method are presented. Instead of specifying the weighting functions, the weighting parameters are used to control the algorithm characteristics. If the numerical solution is approximated by a polynomial of degree n, the approximation is at least nth-order accurate. By choosing the weighting parameters carefully, the order of accuracy can be improved. The generalized PadeH approximations with polynomials of degree n as the numerator and denominator are considered. The weighting parameters are chosen to reproduce the generalized PadeH approximations. Once the weighting parameters are known, any set of linearly independent basic functions can be used to construct the corresponding weighting functions. The stabilizing weighting factions for the weighted residual method are then found explicitly. The accuracy of the particular solution due to excitation is also considered. It is shown that additional weighting parameters may be required to maintain the overall accuracy. The corresponding equations are listed and the additional weighting parameters are solved explicitly. However, it is found that some weighting functions could satisfy the listed equations automatically