On improving mass conservation of level set by reducing spatial discretization errors

The impact of the Level Set (LS) method on numerical capturing of interfaces (Osher and Fedkiw, 2003; Sethian, 1999) is well known. Widely-reaching applications include fluid mechanics, combustion, computer vision, and material science. As a way of describing immiscible multi-phase flows, the LS was shown to provide a worthy alternative (Sussman et al., 1994) to the already in existence volume-tracking (VT) (Noh and Woodward, 1976; Hirt and Nichols, 1981; Youngs, 1982) and front-tracking (FT) (Chern et al., 1985; Unverdi and Tryggvason, 1992) methods. A comparative evaluation, made by means of a set of test problems which represent flows with significant deformation, stretching, and tearing, was presented by Rider and Kothe (1995). In this study, the LS method was shown to lead to loss (or gain) of mass when fluid filaments become comparable to grid size, while the VT methods were found prone to producing ‘‘blobby’’ structures. In other words, both methods exhibited low accuracy in under-resolved flows, while particle-based, front-tracking methods were shown to perform better in conserving filamentary structures. In attempts to improve the mass conservation properties of the LS method a number of modifications have been introduced, including a variety of re-initialization (Sussman et al., 1994; Sussman and Fatemi, 1999) and velocity extension (Adalsteinsson and Sethian, 1999) techniques. While these advancements have been found to be helpful in maintaining the level set as a signed distance function (which is especially important for accurate computation of interface curvature in flows with surface tension), they did not meet the objective for such as the above-mentioned, under-resolved (thin-filament) flows. Recognizing that, several ‘‘hybrid’’ methods have been developed, combining the level set with volume-tracking (Sussman and Puckett, 2000; Sussman, 2003) or front-tracking (Enright et al., 2002, 2005). The method developed by Enright et al. (2002) is particularly interesting. Combining a level set function with Lagrangian marker particles, their Particle Level Set (PLS) method was shown to considerably improve mass conservation in under-resolved flows, while maintaining a smooth geometrical description of the interface. Unfortunately, ‘‘hybrid’’ LS + VT and LS + FT methods become significantly more complicated and computationally expensive, especially in parallel 3D implementations.