In this paper, we propose a spectral projection of a regularized Boussinesq system for wave propagation on the surface of a fluid. The spectral method is based on the use of Legendre polynomials, and is able to handle time-dependent Dirichlet boundary conditions with spectral accuracy. The algorithm is applied to the study of undular bores, and in particular to the onset of wave breaking connected with undular bores. As proposed in [2], an improved version of the breaking criterion recently introduced in [5] is used. This tightened breaking criterion together with a careful choice of the relaxation parameter yields rather accurate predictions of the onset of breaking in the leading wave of an undular bore.