At the magnetopause boundary, the Kelvin-Helmholtz instability should be convective; the excitation of the instability should move tailward with a finite velocity. However previous studies of the Kelvin-Helmholtz instability have been limited to periodic system with system length equal to the initial perturbation wavelength. In this paper the convective Kelvin-Helmholtz instability is modelled by solving the MHD equations numerically as an initial value problem. It is found that the convective Kelvin-Helmholtz instability has the same linear growth rate as the periodic Kelvin-Helmholtz instability. But at the nonlinear stage, the periodic conditions in the periodic system stabilize the nonlinear growth of the instability. In contrast, the convective Kelvin-Helmholtz instability can grow to a much larger amplitude and, thus, introduce many interesting phenomena. For instance, in the perpendicular configuration (B0 ⊥v0), it can create a very large vortex flow and generate shocks off the vortex boundary. In the parallel configuration (B0∥v0), expulsion of the magnetic field is observed when B0 is small, and a boundary layer is formed when B0 is large. Our conclusion is that the Kelvin-Helmholtz instability may be more important than indicated by previous studies. However, it is suggested that its effects on the magnetosphere can be estimated only when realistic geometry and flow conditions are used.