Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xiuhui Guo, Wei Jiang, Chunmei Su

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.

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

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 approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal -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.

Paper Structure

This paper contains 11 sections, 2 theorems, 64 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The LDG method for curve-shortening flow
  3. Basic notations
  4. The LDG method
  5. A fully discrete scheme
  6. Well-posedness
  7. Extension to anisotropic area-preserving curve-shortening flow
  8. Numerical results & discussions
  9. Isotropic case
  10. Anisotropic case
  11. Conclusions

Key Result

theorem 1

Let $(\mu, \boldsymbol{X}, \boldsymbol{q}, \bm{\xi}) \in V_h \times \boldsymbol{W}_h \times \boldsymbol{W}_h \times \boldsymbol{W}_h$ be a solution of the semi-discrete system 2.5-2.6. Then the total discrete energy $W_c^h(t)$, as defined in 2.7, is dissipative during the evolution, i.e., $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Numerical simulation of the CSF starting from an ellipse. Quantitative diagnostics: (a) evolution; (b) relative area loss $\Delta A^h(t)$; (c) mesh ratio $\Psi(t)$; (d) normalized total energy $W^h_c(t)/W_c^h(0)$; (e) normalized energy component $W_1^h(t)/W_c^h(0)$; (f) normalized energy component $W_2^h(t)/W_c^h(0)$.
  • Figure 2: Numerical simulation of the AP-CSF starting from an ellipse with penalty factor $\alpha=1/h$. Quantitative diagnostics: (a) evolution; (b) relative area loss $\Delta A^h(t)$; (c) mesh ratio $\Psi(t)$; (d) normalized total energy $W^h_c(t)/W_c^h(0)$; (e) normalized energy component $W_1^h(t)/W_c^h(0)$; (f) component $W_2^h(t)/W_c^h(0)$.
  • Figure 3: Numerical simulation of the AP-CSF starting from an ellipse with $\alpha=0$: (a) geometric evolution; (b) normalized relative area loss $\Delta A^h(t)$; (c) mesh ratio $\Psi(t)$; (d) normalized total energy $W^h_c(t)/W_c^h(0)$.
  • Figure 4: Numerical simulation of the AP-CSF starting from an ellipse with $\alpha=0.1$: (a) evolution; (b) relative area loss $\Delta A^h(t)$; (c) mesh ratio $\Psi(t)$; (d) normalized total energy $W^h_c(t)/W_c^h(0)$; (e) normalized energy component $W_1^h(t)/W_c^h(0)$; (f) normalized energy component $W_2^h(t)/W_c^h(0)$.
  • Figure 5: Tangential velocity on the initial elliptical curve under the AP-CSF for different numerical fluxes: (a) $\bm{\xi}=\bm{\xi}^+$; (b) $\bm{\xi}=\bm{\xi}^-$; (c) $\bm{\xi}=\bm{\xi}^++\alpha(\boldsymbol{X}^+-\boldsymbol{X}^-)$; (d) $\bm{\xi}=\bm{\xi}^-+\alpha(\boldsymbol{X}^+-\boldsymbol{X}^-)$.
  • ...and 6 more figures

Theorems & Definitions (5)

  • remark 1
  • theorem 1: Energy dissipation
  • proof
  • theorem 2: Well-posedness
  • proof