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.

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  1. Larios-Cárdenas, Luis Ángel; Gibou, Frédéric: A hybrid inference system for improved curvature estimation in the level-set method using machine learning (2022)
  2. Lévy, Bruno: Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries (2022)
  3. Zhang, Qinghai; Li, Zhixuan: Boolean algebra of two-dimensional continua with arbitrarily complex topology (2020)
  4. Prieto, Juan Luis; Carpio, Jaime: A-SLEIPNNIR: a multiscale, anisotropic adaptive, particle level set framework for moving interfaces. Transport equation applications (2019)
  5. Owkes, Mark; Cauble, Eric; Senecal, Jacob; Currie, Robert A.: Importance of curvature evaluation scale for predictive simulations of dynamic gas-liquid interfaces (2018)
  6. Zhang, Qinghai: Fourth- and higher-order interface tracking via mapping and adjusting regular semianalytic sets represented by cubic splines (2018)
  7. Zhang, Qinghai: HFES: a height function method with explicit input and signed output for high-order estimations of curvature and unit vectors of planar curves (2017)
  8. Zhang, Qinghai: GePUP: generic projection and unconstrained PPE for fourth-order solutions of the incompressible Navier-Stokes equations with no-slip boundary conditions (2016)
  9. Zhang, Qinghai; Fogelson, Aaron: MARS: an analytic framework of interface tracking via mapping and adjusting regular semialgebraic sets (2016)
  10. Zhang, Qinghai; Fogelson, Aaron: Fourth-order interface tracking in two dimensions via an improved polygonal area mapping method (2014)