In this paper we present a conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.
Bibliographical noteFunding Information:
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0023794 ). The authors thank the reviewers for the constructive and helpful comments on the revision of this article.
- Cahn-Hilliard equation
- Complex domain
- Dirichlet boundary condition
- Multigrid method
- Unconditionally gradient stable scheme
ASJC Scopus subject areas
- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics