Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with passive convection for given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and fourth finite difference scheme, which is obtained from formulation $Q^2$ spectral element method. prove that discrete maximum principle holds under suitable mesh size step constraints. same result also applies to construct bound-preserving any an