Analysis of high order fast interface tracking methods

• Published in: Numerische Mathematik Vol. 128; no. 2; pp. 339 - 375
• Main Author: Runborg, Olof
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2014
• Subjects: 2-Scale Difference-Equations; Curves; Front-Tracking; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Regularity; Simulation; Smoothness; Subdivision Schemes; Theoretical; Travel-Time; 42C40; 65L20; 65D99
• DOI: 10.1007/s00211-014-0613-5
• ISSN: 0029-599X; 0945-3245

Fast high order methods for the propagation of an interface in a velocity field are constructed and analyzed. The methods are generalizations of the fast interface tracking method proposed in Runborg (Commun Math Sci 7:365–398, 2009 ). They are based on high order subdivision to make a multiresolution decomposition of the interface. Instead of tracking marker points on the interface the related wavelet vectors are tracked. Like the markers they satisfy ordinary differential equations (ODEs), but fine scale wavelets can be tracked with longer timesteps than coarse scale wavelets. This leads to methods with a computational cost of O ( log N / Δ t ) rather than O ( N / Δ t ) for N markers and reference timestep Δ t . These methods are proved to still have the same order of accuracy as the underlying direct ODE solver under a stability condition in terms of the order of the subdivision, the order of the ODE solver and the time step ratio between wavelet levels. In particular it is shown that with a suitable high order subdivision scheme any explicit Runge–Kutta method can be used. Numerical examples supporting the theory are also presented.