High-order Moving Boundary Problems With Adaptive Mesh Refinement

The gradient augmented level set method (GALSM) improves upon the classical level set method by evolving both the level set function and its gradient values in a fully coupled fashion. Advantages include high-order accuracy, reduced volume loss, sub-grid resolution of the interface, and an optimally local advection scheme. Our previous work introduced a narrow-band reinitialization procedure for the GALSM, which was successfully coupled with a simple multiphase Navier-Stokes solver to represent the free surface between water and air. Our ongoing work seeks to couple the GALSM with higher-order numerical methods for the solution of a variety of problems, including the heat equation, Stokes flow, and the Stefan problem in an adaptive mesh refinement (AMR) framework. Preliminary results and ongoing work are presented.