As a sequel to our previous work [Q. Zhang, SIAM J. Numer. Anal., 51 (2013), pp. 2822--2850], [Q. Zhang and A. Fogelson, SIAM J. Sci. Comput., 36 (2014), pp. A2369--A2400], 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 sc...
['Qinghai Zhang', 'Aaron L. Fogelson']
MARS: An Analytic Framework of Interface Tracking via Mapping and Adjusting Regular Semialgebraic Sets