A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension

We take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension. Methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid. However, background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry. Therefore, our incompressibility discretization utilizes a particle based projection near the interface and a background MAC grid based projection for efficiency in the vast interior of the liquid domain - as well as a novel method for coupling these two disparate projections together. We show that the overall coupled elliptic solver is second order accurate, and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems. A similar second order accurate discretization is derived when the MAC grid unknowns are located on faces (as opposed to cell centers) so that Navier-Stokes viscosity can be solved for implicitly as well. Finally, we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step. Beyond stable implicit surface tension for our novel hybrid discretization, we demonstrate preliminary results for both standard front tracking and the particle level set method.