  We consider the interface advection problem by a prescribed velocity field in
the special case when the interface intersects the domain boundary, i.e. in the
presence of a contact line. This problem emerges from the discretization of
continuum models for dynamic wetting. The kinematic evolution equation for the
dynamic contact angle (Fricke et al., 2019) expresses the fundamental
relationship between the rate of change of the contact angle and the structure
of the transporting velocity field. The goal of the present work is to develop
an interface advection method that is consistent with the fundamental
kinematics and transports the contact angle correctly with respect to a
prescribed velocity field. In order to verify the advection method, the
kinematic evolution equation is solved numerically and analytically (for
special cases). We employ the geometrical Volume-of-Fluid (VOF) method on a
structured Cartesian grid to solve the hyperbolic transport equation for the
interface in two spatial dimensions. We introduce generalizations of the Youngs
and ELVIRA methods to reconstruct the interface close to the domain boundary.
Both methods deliver first-order convergent results for the motion of the
contact line. However, the Boundary Youngs method shows strong oscillations in
the numerical contact angle that do not converge with mesh refinement. In
contrast to that, the Boundary ELVIRA method provides linear convergence of the
numerical contact angle transport.
