Rather special initial value problems for systems of second-order differential equations of the type Ÿ(t) = -Ω2Y(t) + G(t,Y(t)) with Y(t0) = Y0 and (t0) = V0 are studied in this paper. It is assumed that Ω is a diagonal matrix of order N and that G is a smooth function (by “smooth,” the authors probably mean that it is continuously differentiable up to some appropriate order, which is required further in the text of the paper, for example in Theorem 2.1). It is additionally assumed that the problem is solved with a time-stepsize h by explicit symmetric cosine multistep methods on an equidistant grid along the time interval.
After a short introductory section, some theoretical results related to the local and global errors of the selected numerical methods are proven in Section 2. Some suggestions for designing methods, which are in some sense optimal, are presented and justified in Section 3.
In Section 4, the phenomenon called resonance is described first for scalar problems and after that for systems of two equations. It is indicated that this phenomenon is sometimes related to the stability of the numerical methods. Filtering techniques are proposed to avoid the appearance of resonance.
Two highly accurate multistep cosine methods (of orders eight and ten, respectively) are explicitly constructed in Section 5. Numerical results are given and discussed in Section 6. Several figures are presented. The authors compare the results they obtained by the two numerical methods they developed with those obtained by two other methods. The results demonstrate the better performance of the methods constructed in this paper.
