A gradient-augmented level set method with an optimally local, coherent advection scheme

TL;DR

This paper introduces a gradient-augmented level set method that uses local, coherent advection schemes to improve surface tracking accuracy and curvature approximation, outperforming traditional methods in stability and detail resolution.

Contribution

The paper presents a novel level set method that incorporates gradient information for fully coupled evolution, achieving high accuracy with local stencils and enhanced structure tracking.

Findings

Comparable quality to WENO schemes

Ability to locate structures smaller than grid size

Accurate curvature approximation using derivatives

Abstract

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.