A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system

Yibao Li, Qian Yu, Weiwei Fang, Binhu Xia, Junseok Kim

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results.

Original languageEnglish
Article number3
JournalAdvances in Computational Mathematics
Issue number1
Publication statusPublished - 2021 Feb

Bibliographical note

Funding Information:
Y.B. Li is supported by the National Natural Science Foundation of China (No. 11601416) and by the China Postdoctoral Science Foundation (No. 2018M640968). The corresponding author (J.S. Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053).

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.


  • Backward differentiation formula
  • Cahn–Hilliard–Hele–Shaw
  • Linear multigrid
  • Second-order accuracy
  • Unique solvability

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system'. Together they form a unique fingerprint.

Cite this