The Multiphase Cubic MARS method for Fourth- and Higher-order Interface Tracking of Two or More Materials with Arbitrarily Complex Topology and Geometry

For interface tracking of an arbitrary number of materials in two dimensions, we propose a multiphase cubic MARS method that (a) accurately and efficiently represents the topology and geometry of the interface via graphs, cycles, and cubic splines, (b) maintains an $(r,h)$-regularity condition of the interface so that the distance between any pair of adjacent markers is within a user-specified range that may vary according to the local curvature, (c) applies to multiple materials with arbitrarily complex topology and geometry, and (d) achieves fourth-, sixth-, and eighth-order accuracy both in time and in space. In particular, all possible types of junctions, which pose challenges to VOF methods and level-set methods, are handled with ease. The fourth- and higher-order convergence rates of the proposed method are proven under the MARS framework. Results of classic benchmark tests confirm the analysis and demonstrate the superior accuracy and efficiency of the proposed method.