Abstract
We discuss how the equations describing the dynamics of an
order parameter based on a continuum theory
should be numerically implemented when rigid, impenetrable
walls are present.
In this study, we pay particular attention to
the cases of non-conserved order parameters.
For concreteness, clarity and simplicity,
we consider a typical example,
the dynamics of a nematic liquid crystal cell
under an in-plane electric field,
where surface anchoring
plays a crucial role.
In the discretized equation of
the dynamics of the order parameter (azimuthal angle) just at the surface,
there appears
a contribution from the bulk energy that is proportional to
the grid spacing and absent
in a usual discretization.
We show that our discretization scheme is consistent
in the sense that it can yield
the identical dynamical behavior irrespective of the
choice of the grid spacing.
We also demonstrate that our scheme is reliable
even when the grid spacing is relatively large.
