In this work, four finite volume methods have been constructed to solve the
2D convective Cahn–Hilliard equation with specified initial condition and periodic
boundary conditions. We prove existence and uniqueness of solutions. The stability
and convergence analysis of the numerical methods have been discussed thoroughly.
The nonlinear terms are approximated by a linear expression based on Mickens’ rule
(Mickens, Nonstandard finite difference models of differential equations. World Scientific,
Singapore, 1994) of nonlocal approximations of nonlinear terms. Numerical
experiments for a test problem have been carried out to test all methods.