  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.
