  The paper is concerned with the numerical approximation of the Intermediate
Long Wave and Benjamin-Ono systems, that serve as models for the propagation of
interfacial internal waves in a two-layer fluid system in particular physical
regimes. The paper focuses on two issues of approximation. First, the spectral
Fourier-Galerkin method is used to discretize in space the corresponding
periodic initial-value problems, and the error of the semidiscretizations is
analyzed. The second issue concerns the numerical generation of solitary-wave
solutions of the systems. We use acceleration techniques to improve the
computation of the approximate solitary waves and check their performance with
numerical examples.
