A stable finite difference method for a Cahn-Hilliard type equation with long-range interaction

We propose a stable numerical scheme for a Cahn-Hilliard type equation with long-range interaction describing the micro-phase separation of diblock copolymer melts. The scheme is designed by using the discrete variational derivative method, one of structure preserving numerical methods. The derivation of the discrete variational derivative of a discretized energy functional is simplified by using a suitable discrete L2 space and fractional powers of a discrete approximation of the Laplace operator. The proposed scheme has the same characteristic properties, mass conservation and energy dissipation, as the original equation does. We also discuss the stability and unique solvability of the scheme.