A p-adaptive local discontinuous galerkin level set method for Willmore flow

TL;DR

This paper introduces a p-adaptive local discontinuous Galerkin level set method for simulating Willmore flow, combining energy stability, mass conservation, and efficiency improvements through adaptive polynomial degrees and semi-implicit time stepping.

Contribution

It develops a novel p-adaptive LDG level set approach with a semi-implicit Runge-Kutta scheme and multi-grid solver for efficient Willmore flow simulation.

Findings

The method is energy stable and mass conservative.

Adaptive polynomial degrees improve computational efficiency.

Numerical examples demonstrate the method's effectiveness.

Abstract

The level set method is often used to capture interface behavior in two or three dimensions. In this paper, we present a combination of local discontinuous Galerkin (LDG) method and level set method for simulating Willmore flow. The LDG scheme is energy stable and mass conservative, which are good properties comparing with other numerical methods. In addition, to enhance the efficiency of the proposed LDG scheme and level set method, we employ a p-adaptive local discontinuous Galerkin technique, which applies high order polynomial approximations around the zero level set and low order ones away from the zero level set. A major advantage of the level set method is that the topological changes are well defined and easily performed. In particular, given the stiffness of Willmore flow, a high order semi-implicit Runge-Kutta method is employed for time discretization, which allows larger time step. The equations at the implicit time level are linear, we demonstrate an efficient and practical multi-grid solver to solve the equations. Numerical examples are given to illustrate the combination of the LDG scheme and level set method provides an efficient and practical approach when simulating the Willmore flow.