  The convective Allen-Cahn equation has been widely used to simulate
multi-phase flows in many phase-field models. As a generalized form of the
classic Allen-Cahn equation, the convective Allen-Cahn equation still preserves
the maximum bound principle (MBP) in the sense that the time-dependent solution
of the equation with appropriate initial and boundary conditions preserves for
all time a uniform pointwise bound in absolute value. In this paper, we develop
efficient first- and second-order exponential time differencing (ETD) schemes
combined with the linear stabilizing technique to preserve the MBP
unconditionally in the discrete setting. The space discretization is done using
the upwind difference scheme for the convective term and the central difference
scheme for the diffusion term, and both the mobility and nonlinear terms are
approximated through the linear convex interpolation. The unconditional
preservation of the MBP of the proposed schemes is proven, and their
convergence analysis is presented. Various numerical experiments in two and
three dimensions are also carried out to verify the theoretical results.
