On boundary condition capturing for multiphase interfaces

This review paper begins with an overview of the boundary condition capturing approach to solving problems with interfaces. Although the authors' original motivation was to extend the ghost fluid method from compressible to incompressible flow, the elliptic nature of incompressible flow quickly quenched the idea that ghost cells could be defined and used in the usual manner. Instead the boundary conditions had to be implicitly captured by the matrix formulation itself, leading to the novel approach. We first review the work on the variable coefficient Poisson equation, noting that the simplicity of the method allowed for an elegant convergence proof. Simplicity and robustness also allowed for a quick extension to three-dimensional two-phase incompressible flows including the effects of viscosity and surface tension, which is discussed subsequently. The method has enjoyed popularity in both computational physics and computer graphics, and we show some comparisons with the traditional delta function approach for the visual simulation of bubbles. Finally, we discuss extensions to problems where the velocity is discontinuous as well, as is the case for premixed flames, and show an example of multiple interacting liquids that includes all of the aforementioned phenomena.