Oscillation-free advection of interfaces with high order semi-Lagrangian schemes

High order interpolation methods improve the accuracy of semi-Lagrangian advection, however, mostly due to oscillations, only up to third order methods have been used. B-splines are known to have good qualities in terms of accuracy, resolution, and conservation and these properties improve as the order increases. However, high orders suffer from oscillations. In this paper, we propose the use of high order B-splines in interface advection and suggest that an appropriate interface description can improve the accuracy of the advection method and reduce dissipation, dispersion and mass errors while removing the oscillations. Two different interface models are considered. First one is the well-known Cahn–Hilliard equation [Cahn JW, Hilliard JE. Free energy of a nonuniform system, interfacial free energy. J Chem Phys 1958;28:258–67] of diffuse interfaces. We found that the energy minimization property of the Cahn–Hilliard dynamics can also be useful in removing the oscillations and obtaining a monotone solution in semi-Lagrangian advection. The second model is the artificial surface tension method introduced by Yabe et al. [Yabe T, Xiao F. Description of complex and sharp interface during shock wave interaction with liquid drop. J Phys Soc Jpn 1993;62(8):2537–40]. Although it does not provide a physically realistic interface description and as accurate a correction for oscillations, it can be advantageous in tracking high resolution advection lines with high order B-splines. Another drawback of B-splines is their computational complexity. We used a fast, stable algorithm proposed by Unser et al. [Unser M, Aldroubi A, Eden M. B-spline signal processing: part I – theory. IEEE Trans Signal Process 1993;41(2):821–32; Unser M, Aldroubi A, Eden M. B-spline signal processing: part II – efficient design and applications. IEEE Trans Signal Process 1993;41(2):834–48; Unser M. Splines: a perfect fit for signal and image processing. IEEE Signal Process Magaz 1999;(6):22–38] in signal processing. This method is based on a digital filter representation of splines and eliminates the expensive matrix solutions. This algorithm has not been used in semi-Lagrangian methods or more generally in computational fluid dynamics (CFD) to our knowledge.