On a Family of Unsplit Advection Algorithms for Volume-of-Fluid Methods

Volume-of-fluid (VOF) methods are widely-used for the interface tracking problem, yet rigorous analyses of them are rare. This paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations, differential geometry, and Boolean algebra. Based on the concept of donating region (DR) [Q. Zhang, SIAM Rev., 55 (2013), pp. 443--461], the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function. One edgewise solution provides a unified view of DR-based advection algorithms for VOF methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method. The well-known second-order convergence rates of streamline-based VOF advection algorithms are proved rigorously. Potential deterioration of this second-order convergence is also discussed for several subtle issues such as exact mass conservation a...