We examine the use of orthogonal spline collocation for the semidiscretization of the cubic Schrödinger equation and the two- dimensional parabolic equation of F. D. Tappert
[The parabolic method. Lect. Notes Physics 70, 224-287 (1977; Zbl 0399.76079
)]. In each case, an optimal order
estimate of the error in the semidiscrete approximation is derived. For the cubic Schrödinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library.