  Droplets move on substrates with a spatio-temporal wettability pattern as
generated, for example, on light-switchable surfaces. To study such cases, we
implement the boundary-element method to solve the governing Stokes equations
for the fluid flow field inside and on the surface of a droplet and supplement
it by the Cox-Voinov law for the dynamics of the contact line. Our approach
reproduces the relaxation of an axisymmetric droplet in experiments, which we
initiate by instantaneously switching the uniform wettability of a substrate
quantified by the equilibrium contact angle. In a step profile of wettability
the droplet moves towards higher wettability. Using a feedback loop to keep the
distance or offset between step and droplet center constant, induces a constant
velocity with which the droplet surfes on the wettability step. We analyze the
velocity in terms of droplet offset and step width for typical wetting
parameters. Moving instead the wettability step with constant speed, we
determine the maximally possible droplet velocities under various conditions.
The observed droplet speeds agree with the values from the feedback study for
the same positive droplet offset.
