On the accuracy of Galerkin methods in the time domain

In this article, the solution accuracy of the time Galerkin methods is studied. The Bubnov-Galerkin method and the Petrov-Galerkin method are considered. The second-order differential equations are expressed in the first-order form before the Galerkin methods are applied. It is well known that the orders of accuracy for the Bubnov-Galerkin method and the Petrov-Galerkin method are m and 2m, respectively, at the end of a time interval if polynomials with m undetermined coefficients are used as interpolation functions in the time interval. In this article, the accuracy of the interpolated solutions within the time interval is investigated by making a comparison with the exact solutions. It is found that the order of accuracy for both the Bubnov-Galerkin method and the Petrov-Galerkin method is, in general, m only within the time interval under consideration. It is also found that there are some locations with one order higher in accuracy. Besides, it is shown that for the Petrov-Galerkin method to maintain higher order accuracy at the end of the time interval, the excitation should be accurate up to order 2m.