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

Abstract

We present an improved PAM (iPAM) method as the first fourth-order interface tracking method whose convergence rates are independent of $C^1$ (derivative) discontinuities of the interface. As an improved version of the polygonal area mapping (PAM) method [Q. Zhang and P. L.-F. Liu, J. Comput. Phys., 227 (2008), pp. 4063--4088], the accuracy of the iPAM method is achieved via (i) augmenting the abstract data structure of PAM to faithfully represent multiple components of material regions within a single cell, (ii) removing restrictive assumptions of PAM, (iii) adjusting the volume of represented cell material regions via polygon ear removal, and (iv) maintaining a relation $(h_L=r_hh^{\alpha})$ between the Eulerian grid size $h$ and the Lagrangian length scale that measures the distance between adjacent interface markers. Unlike volume-of-fluid methods and level-set methods, merging and separation of the tracked material is not automatically handled by the iPAM method. Instead, flexible subgrid resolutions...