  In this paper, a fractional step lattice Boltzmann method is proposed to
model two-phase flows with large density differences by solving Cahn-Hilliard
phase-field equation and the incompressible Navier-Stokes equations.In order to
maintain a hyperbolic tangent property of the interface profile and conserve
the volume, an interfacial profile correction term and a flux correction term
are added into the original Cahn-Hilliard equation respectively. By using a
fractional step scheme, the modified Cahn-Hilliard equation is split into two
sub-equations. One is solved in the framework of lattice Boltzmann equation
method. The other is solved by the finite difference method. Compared with the
previous lattice Boltzmann methods, the proposed method is able to maintain the
order parameter within a physically meaningful range, which is conductive to
track the interface accurately. In addition, the multi-relaxation-time
collision model and a high-order compact selective filter operation are
employed to enhance the numerical stability. The proposed method can simulate
two-phase fluid flows with the density ratio up to $1000$. In order to validate
the accuracy and capability of the method, several benchmark problems,
including single vortex deform of a circle, translation of a drop,
Laplace-Young law, capillary wave and rising bubble with large density ratios,
are presented. The results are in good agreement with the analytical solutions
and the data in the literature for the investigated benchmarks.
