  We propose a method for the numerical solution of a multiscale model
describing sorption kinetics of a surfactant around an oscillating bubble. The
evolution of the particles is governed by a convection-diffusion equation for
the surfactant concentration $c$, with suitable boundary condition on the
bubble surface, which models the action of the short range attractive-repulsive
potential acting on them when they get sufficiently close to the surface
\cite{multiscale_mod}. In the domain occupied by the fluid, the particles are
transported by the fluid motion generated by the bubble oscillations. The
method adopted to solve the equation for $c$ is based on a finite-difference
scheme on a uniform Cartesian grid and implemented in 2D and 3D axisymmetric
domains. We use a level-set function to define the region occupied by the
bubble, while the boundary conditions are discretized by a ghost-point
technique to guarantee second order accuracy at the curved boundary. The sparse
linear system is finally solved with a geometric multigrid technique designed
\textit{ad-hoc\/} for this specific problem. Several accuracy tests are
provided to prove second order accuracy in space and time.
  The fluid dynamics generated by the oscillating bubble is governed by the
Stokes equation solved with a second order accurate method based on a
monolithic approach, where the momentum and continuity equations are solved
simultaneously. Since the amplitude of the bubble oscillations are very small,
a simplified model is presented where the computational bubble is actually
steady and its oscillations are represented purely with time-dependent boundary
conditions. A numerical comparison with the moving domain model confirms that
this simplification is perfectly reasonable for the class of problems
investigated in this paper.
