A locally gradient-preserving reinitialization for level set functions

TL;DR

This paper introduces a new reinitialization method for level set functions that preserves interface gradients, improving accuracy in physical simulations involving interface stretching.

Contribution

The paper develops a locally gradient-preserving reinitialization (LGPR) method involving three PDEs, with proofs of well-posedness and high-accuracy numerical schemes.

Findings

Preserves interface gradient during reinitialization.

Uses high-accuracy subcell resolution in numerical solutions.

Computationally efficient for small interface bands.

Abstract

The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. The gradient of the level set function represents the stretching of the interface, which is of critical importance in many physical applications. The proposed locally gradient-preserving reinitialization (LGPR) method involves the solution of three PDEs of Hamilton-Jacobi type in succession; first the signed distance function is found using a traditional reinitialization technique, then the interface gradient is extended into the domain by a transport equation, and finally the new level set function is achieved with the solution of a generalized reinitialization equation. We prove the well-posedness of the Hamilton-Jacobi equations, with possibly discontinuous Hamiltonians, and propose numerical schemes for their solutions. A subcell resolution technique is used in the numerical solution of the transport equation to extend data away from the interface directly with high accuracy. The reinitialization technique is computationally inexpensive if the PDEs are solved only in a small band surrounding the interface. As an important application, the LGPR method will enable the application of the local level set approach to the Eulerian Immersed boundary method.