Stabilized Radial Basis Function Finite Difference Schemes with Mass Conservation for the Cahn-Hilliard Equation on...

Area of Science:

• Computational physics
• Materials science
• Numerical analysis

Background:

• The Cahn-Hilliard equation governs phase separation and conserves mass.
• Radial basis function-finite difference (RBF-FD) methods excel at complex geometries.
• Existing RBF-FD schemes struggle with mass conservation on closed surfaces.

Purpose of the Study:

• To develop mass-conservative RBF-FD schemes for the Cahn-Hilliard equation on smooth closed surfaces.
• To ensure numerical solutions accurately preserve the mass conservation property.
• To enable efficient long-term simulations of phase separation.

Main Methods:

• Spatial discretization using RBF-FD.
• Temporal integration with Backward Euler (BDF1) and Crank-Nicolson (CN) schemes.
• Incorporation of an L2 projection method for mass conservation.

Main Results:

• Proposed RBF-FD schemes successfully preserve mass conservation.
• The methods are effective under relatively mild time-step constraints.
• Numerical experiments validate the schemes' performance for long-term simulations.

Conclusions:

• The developed RBF-FD schemes provide accurate and mass-conservative solutions for the Cahn-Hilliard equation on surfaces.
• These methods overcome limitations of existing approaches regarding mass conservation.
• The schemes facilitate efficient and reliable long-term phase-separation modeling.