Comparison study of the conservative Allen–Cahn and the Cahn–Hilliard equations

•We compare conservative phase-field models.•Generation and motion of interface are studied.•Geometric flows are given and matched to the phase-field equations.•Modified Allen–Cahn equation is solved using a Lagrange multiplier method.

In this paper, a comparison study of conservative Allen–Cahn and Cahn–Hilliard equations is presented. We consider two mass-conservative Allen–Cahn equations and two Cahn–Hilliard equations with constant and variable mobilities. The equations are discretized using finite difference schemes, and discrete systems of the equations are solved using a nonlinear multigrid method. The generation and motion of interface are investigated for the conservative equations. We then present numerical experiments which highlight different dynamics of the four equations.