An integral equation method for the Cahn-Hilliard equation in the wetting problem

Xiaoyu Wei, Shidong Jiang, Andreas Klöckner, Xiao Ping Wang

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4 Scopus citations


We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, the source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension.

Original languageEnglish (US)
Article number109521
JournalJournal of Computational Physics
StatePublished - Oct 15 2020

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


  • Cahn-Hilliard equation
  • Convex splitting
  • Integral equation method
  • Second-kind integral equation
  • Volume potential
  • Young's angle


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