  The modeling and simulation of two-phase flows is still an active research
area, mainly when surface tension is present. One way to model the different
phases is with interface capturing methods. Two well-established interface
capturing approaches are the level-set and phase-field methods. The level-set
method is known for its ability to compute the surface tension accurately, and
phase-field models satisfy the second law of thermodynamics. This paper reviews
and compares a level-set and a phase-field approach to simulate two-phase
flows. We use a modified level-set method, called convected level-set. The
difference to the standard level-set method is that the re-initialization step
is embedded in the convection equation, avoiding a separate step during the
calculation. We also apply a global mass conservation procedure to enforce the
mass balance between phases. On the other hand, the phase-field approach uses a
conservative Allen-Cahn equation with a Lagrange multiplier to conserve the
mass. The methods are implemented in libMesh, a parallel adaptive finite
element library, using the same finite element formulations, time-marching
schemes, solvers, and mesh adaptivity strategies. We present numerical
solutions for the two methods considering adaptive mesh refinement and
coarsening (AMR/C). Results are analyzed and discussed.
