We numerically investigate the interaction between multi-compactons of the K(m,n,p) equation by a finite difference scheme that is of the second-order accuracy and absolutely stable in linearization sense. By adding an artificial dissipation term, it works well for preventing the break-up phenomena of the numerical solutions. Firstly, we simulate the long-time evolution behaviors of the single-compacton to verify the validity of the numerical method. It is shown that the numerical method is effective for solving this problem. Secondly, we study the nonlinear interaction between two-compacton and three-compacton by this numerical method. The numerical results indicate that the wave-frame and wave-velocity after collision are nearly the same as before collision. However, compacton-anticompacton pair induced behind the wave arises with small amplitudes.