  For capillary driven flow the interface curvature is essential in the
modelling of surface tension via the imposition of the Young--Laplace jump
condition. We show that traditional geometric volume of fluid (VOF) methods,
that are based on a piecewise linear approximation of the interface, do not
lead to an interface curvature which is convergent under mesh refinement in
time-dependent problems. Instead, we propose to use a piecewise parabolic
approximation of the interface, resulting in a class of piecewise parabolic
interface calculation (PPIC) methods. In particular, we introduce the parabolic
LVIRA and MOF methods, PLVIRA and PMOF, respectively. We show that a Lagrangian
remapping method is sufficiently accurate for the advection of such a parabolic
interface. It is numerically demonstrated that the newly proposed PPIC methods
result in an increase of reconstruction accuracy by one order, convergence of
the interface curvature in time-dependent advection problems and Weber number
independent convergence of a droplet translation problem, where the advection
method is coupled to a two-phase Navier--Stokes solver. The PLVIRA method is
applied to the simulation of a 2D rising bubble, which shows good agreement to
a reference solution.
