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

TL;DR

This paper develops energy-stable local discontinuous Galerkin methods for solving second and fourth order PDEs on surfaces, validated through numerical experiments on static 2D manifolds.

Contribution

It introduces new second-order accurate, energy-stable DG schemes for surface PDEs with detailed design and validation.

Findings

Schemes are second-order accurate on triangulated surfaces.

Numerical experiments confirm stability and accuracy.

Methods are applicable to second and fourth order PDEs on surfaces.

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.