MARS: An Analytic Framework of Interface Tracking via Mapping and Adjusting Regular Semialgebraic Sets

Abstract

, 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.

Fourth-Order Interface Tracking in Two Dimensions via an Improved Polygonal Area Mapping Method

“…We state the following conclusion and refer the reader to [33] for a proof. It follows that setting α = 3 2 yields a third-order method and α = 2 a fourth-order method in terms of the Eulerian grid size h. This is confirmed by numerical results in section 5.…”

“…In[33], the representation error is defined asE REP := φ T −t 0 t 0 (M(t 0 )) ⊕ φ T −t 0 t 0 (M 0 ) ,where φ is the exact flow map. In incompressible flows, the total volume of the material region remains constant.…”

A high-order fictitious-domain method for the advection-diffusion equation on time-varying domain

“…In the immersed boundary method, the moving boundary is represented by line segments of interface markers and tracked by a fronting tracking method. As such, the overall accuracy is at best second-order [33]. For IIMs, there exist fourth-order schemes for stationary interfaces [21], but we are not aware of any fourthorder IIMs capable of handling nontrivial deforming boundaries.…”

“…We answer this need by developing in this work third-order and fourth-order methods for numerically solving the advection-diffusion equation (2.1) on time-varying domains. For the moving boundaries, we track their loci to fourth-order accuracy by the recent cubic MARS method [35], which features topological modeling of fluids [36] and a rigorous analytic framework for error estimates [33]. For spatial discretization, we adopt a high-order fictitious-domain method on a fixed finite element mesh that covers the full movement range of the deforming domain.…”

High-order finite element methods for nonlinear convection-diffusion equation on time-varying domain

“…We refer to [3,16,17,22,42] and the reference therein for level set methods, to [21,23,28] for volume-of-fluid methods, to [1,2,15] for momentof-fluid methods, and to [47,54] for front-tracking method. In a series of works [50][51][52][53], Zhang developed the cubic MARS algorithm for tracking the loci of free surface. The algorithm traces control points on the surface and forms an approximate surface by cubic spline interpolations.…”