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Local Discontinuous Galerkin Methods for Solving Convection-Diffusion and Cahn-Hilliard Equations on Surfaces

Shixin Xu, Zhiliang Xu

TL;DR

The paper develops Local Discontinuous Galerkin methods for time-dependent convection-diffusion and Cahn–Hilliard equations posed on static surfaces, using a polyhedral approximation Γ_h and carefully designed numerical fluxes to ensure energy stability. By reformulating the equations as first-order systems and employing TVD Runge–Kutta time stepping, the authors obtain schemes that are provably energy-stable and converge with second-order accuracy on planar surface meshes. A comprehensive set of numerical experiments on spheres and other surfaces confirms the method's stability and accuracy, and CH simulations illustrate pattern formation on complex geometries. The work advances intrinsic surface DG methods, highlighting practical performance and identifying avenues for higher-order curved-element discretizations and curvature-aware CFL considerations.

Abstract

Local discontinuous Galerkin methods are developed for solving second order and fourth order time-dependent partial differential equations defined on static 2D manifolds. These schemes are second-order accurate with surfaces triangulized by planar triangles and careful design of numerical fluxes. The schemes are proven to be energy stable. Various numerical experiments are provided to validate the new schemes.

Local Discontinuous Galerkin Methods for Solving Convection-Diffusion and Cahn-Hilliard Equations on Surfaces

TL;DR

The paper develops Local Discontinuous Galerkin methods for time-dependent convection-diffusion and Cahn–Hilliard equations posed on static surfaces, using a polyhedral approximation Γ_h and carefully designed numerical fluxes to ensure energy stability. By reformulating the equations as first-order systems and employing TVD Runge–Kutta time stepping, the authors obtain schemes that are provably energy-stable and converge with second-order accuracy on planar surface meshes. A comprehensive set of numerical experiments on spheres and other surfaces confirms the method's stability and accuracy, and CH simulations illustrate pattern formation on complex geometries. The work advances intrinsic surface DG methods, highlighting practical performance and identifying avenues for higher-order curved-element discretizations and curvature-aware CFL considerations.

Abstract

Local discontinuous Galerkin methods are developed for solving second order and fourth order time-dependent partial differential equations defined on static 2D manifolds. These schemes are second-order accurate with surfaces triangulized by planar triangles and careful design of numerical fluxes. The schemes are proven to be energy stable. Various numerical experiments are provided to validate the new schemes.
Paper Structure (14 sections, 5 theorems, 58 equations, 4 figures, 3 tables)

This paper contains 14 sections, 5 theorems, 58 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model Problems and Preliminaries
  3. Numerical Schemes
  4. LDG Method for Surface Convection-Diffusion Equation
  5. LDG Method for Cahn-Hilliard Equation
  6. Time discretization
  7. Numerical results
  8. Linear advection on a sphere
  9. Diffusion on a sphere
  10. Convection-diffusion on a sphere
  11. Cahn-Hilliard equation on surfaces
  12. Conclusion
  13. Acknowledgement
  14. Nonlinear Surface Transport

Key Result

Lemma 1

(AntDed15) Let $v\in H^1({\Gamma}_h)$ and $\mathbf{r} \in [H^1({\Gamma}_h)]^3$. Then

Figures (4)

  • Figure 1: Two elements ${\mathbf{K}}^{e,+}_{h}$ and ${\mathbf{K}}^{e,-}_{h}$ and their respective conormals $\mathbf{n}^+_h$ and $\mathbf{n}^-_h$ on the common edge $e$.
  • Figure 2: Numerical solutions of the concentration $u$ at time $t = 0.0, 0.1, 0.2, 0.3$ on unit sphere $\mathbb{S}_1$ with 2964 nodes and 5924 triangles.
  • Figure 3: Numerical solutions of the concentration $u$ at time $t = 0.0, 0.1, 0.2$ on ellipsoid $\mathbb{S}_2$ triangulated with 6374 nodes and 12744 triangles.
  • Figure 4: Numerical solutions of the concentration $u$ at time $t = 0.0, 0.1, 0.2, 0.3$ on biconcave surface $\mathbb{S}_3$ with 3904 nodes and 7804 triangles.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proposition 1: $L^2$-stability of LDG scheme (\ref{['eq:ldg']})
  • proof
  • proposition 2: energy stability of LDG scheme (\ref{['eq:ldg_CH']})
  • proof
  • Remark 1
  • Lemma 2
  • ...and 2 more