Pseudospectral vs. finite difference methods for initial value problems with discontinuous coefficients

An initial value problem with piecewise-constant coefficients is considered. The accuracies for both finite difference methods and the pseudospectral method are analyzed, and a modification of the initial value problem is suggested. The modified problem is shown to have the same temporal period as the original problem does, and a second-order accuracy is obtained for the pseudospectral method at integral multiples of the temporal period.