MARS

MARS: an analytic framework of interface tracking via mapping and adjusting regular semialgebraic sets. As a sequel to our previous work [SIAM J. Numer. Anal. 51, No. 5, 2822--2850 (2013; Zbl 1282.65113); SIAM J. Sci. Comput. 36, No. 5, A2369-A2400 (2014; Zbl 06390284)], this paper presents MARS, a generic framework for analyzing interface tracking (IT) methods via mapping and adjusting regular semialgebraic sets. Our mathematical model for moving material regions is the metric space of bounded regular semianalytic sets, equipped with Boolean algebras and advected by homeomorphic flow maps of a nonautonomous ordinary differential equation. By examining the actions of semidiscrete and discrete flow maps upon this metric space, we pinpoint in Lemma 3.9 a fundamental difficulty in achieving an IT accuracy higher than the second order. We then propose a generic IT method by concatenating three unitary operations on the modeling space, bound its overall IT error by the sum of intuitively defined error terms, and further estimate the individual errors in terms of the time step size and a Lagrangian length scale. The analytic utility of MARS is demonstrated by applying it to analyze a variety of IT methods, including volume-of-fluid (VOF) methods and the improved PAM method (iPAM). MARS has a great potential in helping the further development of highly accurate and efficient IT methods. As an example, a cubic iPAM method inspired by MARS resolves two vortex-shear tests to machine precision on a 128-by-128 grid; it could also be vastly superior to VOF methods in terms of efficiency. Fourth-order accuracy in curvature estimation is also achieved under the framework of MARS.