High-order parametric local discontinuous Galerkin methods for anisotropic curve-shortening flows

TL;DR

This paper introduces high-order LDG methods with a parametric approach and semi-implicit time discretization for anisotropic curve-shortening flows, demonstrating stability, high accuracy, and robustness against mesh degradation.

Contribution

The paper develops a novel high-order LDG scheme that is unconditionally energy dissipative and stable on poor meshes, outperforming classical methods in anisotropic geometric flow simulations.

Findings

Achieves optimal $(k+1)$-order accuracy for $P^k$ approximations.

Proves unconditional energy dissipation for the semi-discrete scheme.

Remains stable on severely degraded meshes, capturing complex geometric evolutions.

Abstract

We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For $P^k$ approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal $(k+1)$-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.