  Central finite difference schemes have long been avoided in the context of
two-phase flows for the advection of the phase indicator function due to
numerical overshoots and undershoots associated with their dispersion errors.
We will show however, for an incompressible flow, in the context of a specific
diffuse interface model, one can maintain the boundedness of the phase field
while also taking advantage of the low cost and ease of implementation of
central differences to construct a non-dissipative discretization scheme for
the advective terms. This is made possible by combining the advection and
reinitialization steps of a conservative level set scheme introduced by Olsson
and Kreiss [J. Comput. Phys., 210, 225 (2005)] to form a phase field equation
similar to that of Chiu and Lin [J. Comput. Phys., 230, 185 (2011)]. Instead of
resorting to specialized upwind methods as in these articles, we prove that the
boundedness of the phase field is guaranteed for certain choices of the free
parameters ($\epsilon$ and $\gamma$) for a specific central difference scheme
that we propose. The proposed discretely conservative and bounded phase field
equation, which is free of any reinitialization or mass redistribution,
possesses desirable properties that can be leveraged in the coupled finite
difference discretization of the two-phase momentum equation. Additionally, as
compared to the state-of-the-art conservative and bounded two-phase flow
methods, the proposed method boasts competitive accuracy-vs-cost trade-off,
small memory requirements, ease of implementation and excellent
parallelizability, providing a viable alternative for realistic two-phase flow
calculations.
