A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

Authors

  • Darae Jeong Department of Mathematics, Kangwon National University, Gangwon-do 24341, Republic of Korea
  • Yibao Li School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
  • Chaeyoung Lee Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
  • Junxiang Yang Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
  • Junseok Kim Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

DOI:

https://doi.org/10.4208/cicp.OA-2018-0202

Keywords:

Cahn–Hilliard equation, mass correction scheme, narrow band domain, closest point method.

Abstract

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.

Published

2019-12-07

Issue

Section

Articles