In this work, we analyze four finite volume methods for the nonlinear convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The methods used are: implicit one-level, explicit one-level, implicit multilevel and explicit multilevel finite volume methods. The existence and uniqueness of solution, convergence and stability of the finite volume solutions are proved. We compute L2- error and rate of convergence for all methods. We then compare the multilevel methods with the one-level methods by means of stability and CPU time. It is shown that the multilevel finite volume method is faster than the one-level method.