A new numerical scheme for computing the evolution of water waves with a moderate curvature of the free surface, modeled by the dispersive shallow water equations, is described. The discretization of this system of equations is faced with two kinds of numerical difficulties: the nonsymmetric character of the (nonlinear) advectionpropagation operator and the presence of third order mixed derivatives accounting for the dispersion phenomenon. In this paper it is shown that the Taylor-Galerkin finite element method can be used to discretize the problem, ensuring second order accuracy both in time and space and guaranteeing at the same time unconditional stability. The properties of the scheme are investigated by performing a numerical stability analysis of a linearized model of the scalar 1D regularized long wave equation. The proposed scheme extends straightforwardly to the fully nonlinear 2D system, which is solved here for the first time on arbitrary unstructured meshes. The results of the numerical simulation of a solitary wave overpassing a vertical circular cylinder are presented and discussed in a physical perspective.