This paper is concerned with solving the viscous and inviscid shallow water equations. The numerical method is based on second-order finite volume-finite element (FV-FE) discretization: the convective inviscid terms of the shallow water equations are computed by a finite volume method, while the diffusive viscous terms are computed with a finite element method. The method is implemented on unstructured meshes. The inviscid fluxes are evaluated with the approximate Riemann solver coupled with a second-order upwind reconstruction. Herein, the Roe and the Osher approximate Riemann solvers are used respectively and a comparison between them is made. Appropriate limiters are used to suppress spurious oscillations and the performance of three different limiters is assessed. Moreover, the second-order conforming piecewise linear finite elements are used. The second-order TVD Runge-Kutta method is applied to the time integration. Verification of the method for the full viscous system and the inviscid equations is carried out. By solving an advection-diffusion problem, the performance assessment for the FV-FE method, the full finite volume method, and the discontinuous Galerkin method is presented.