Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation

Abstract

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

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

“…The proposed scheme is unconditionally energy-stable by choosing proper parameters. Shen and Zhang [26] considered a high-order finite difference scheme for a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. They proved that the discrete maximum principle holds under suitable mesh size and time step constraints.…”

Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows

“…Apart from the central difference discretization discussed above, the lumped mass finite element method with piecewise linear basis functions can also be adopted and Lemma 2.3 still holds correspondingly [14]. In addition, there have been some initial explorations on the MBP-preserving methods using the fourth-order accurate spatial discretization, such as the compact difference approximation [49] and the finite difference formulation of the Q 2 spectral element method [47], combined with the Euler-type time-stepping approaches. However, it is not obvious that whether these fourth-order discrete Laplace operators satisfy Lemma 2.3, and thus it is worthy of further investigations on the combination of higher-order spatial discretizations with the exponential integrator methods studied in this paper.…”