Abstract: For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations, we show that the simple bound-preserving limiter in Li et al. (SIAM J Numer Anal 56: 3308-3345, 2018) can enforce the strict bounds of the vorticity, if the velocity field satisfies a discrete divergence free constraint. For reducing oscillations, a modified TVB limiter adapted from Cockburn and Shu (SIAM J Numer Anal 31: 607-627, 1994) is constructed without affecting the bound-preserving property. This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.

Key words: Finite difference, Monotonicity, Bound-preserving, Discrete maximum principle, Passive convection, Incompressible flow, Total variation bounded limiter