ARMS: Adding and removing markers on splines for high-order general interface tracking under the MARS framework

Abstract

Based on the MARS framework for interface tracking (IT) (Zhang and Fogelson (2016) [33]) (Zhang and Li (2020) [35]), we propose ARMS (adding and removing markers on splines as a strategy for regularizing distances between adjacent markers in simulating moving boundary problems. In ARMS, we represent the interface by cubic/quintic splines and add/remove interface markers at each time step to maintain a roughly uniform distribution of chordal lengths. To demonstrate the utility of ARMS, we apply it to two-dimensional mean curvature flows to develop ARMS-MCF2D, where spatial derivatives are approximated by finite difference formulas and the resulting nonlinear system of ordinary differential equations is solved by explicit, semi-implicit, and implicit Runge–Kutta methods. As such, the semi-implicit and implicit ARMS-MCF2D methods are unconditionally stable. Error analysis indicates that the order of accuracy of ARMS-MCF2D can be 2, 4, or 6. Results of numerical experiments confirm the analysis, show the effectiveness of ARMS in maintaining the regularity of interface markers, and demonstrate the superior accuracy of ARMS-MCF2D over other existing methods. A one-phase Stefan problem is also simulated by coupling ARMS to a finite difference method, exhibiting its potential utility to general IT in moving boundary problems. The generality and flexibility of ARMS partially lie in the fact that, by specifying a discrete time integrator for her own moving boundary problem, an application scientist immediately obtains a MARS method for the problem without worrying about curve fitting and marker distributions.