Development and application of a parallel multigrid solver for the simulation of spreading droplets

We consider the parallel application of an efficient solver developed for the accurate solution of a range of droplet spreading flows modelled as a coupled set of nonlinear lubrication equations. The underlying numerical scheme is based upon a second order finite difference discretization in space and a second order, fully implicit, adaptive scheme in time. At each time step this leads to the need to solve a large system of nonlinear algebraic equations, for which the full approximation storage (FAS) multigrid algorithm is employed. The motion of the contact line between the three phases (liquid, air and the solid substrate) is based upon the assumption of a thin precursor film, with a corresponding disjoining pressure term in the governing equations. It is the inclusion of this precursor film in the model that motivates the need for a parallel solution method. This is because the thickness of such a film must be very small in order to yield realistic predictions whilst the finite difference grid must be correspondingly fine in order to obtain accurate numerical solutions. Results are presented which demonstrate that the parallel implementation is sufficiently efficient and robust to allow reliable numerical solutions to be obtained for a level of mesh resolution that is an order of magnitude finer than is possible using a single processor. Copyright c 2007 John Wiley & Sons, Ltd.