Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface...

Area of Science:

• Computational physics
• Mathematical modeling
• Biomedical engineering

Background:

• Diffuse-interface models are crucial for simulating phase transitions and biological processes like tumor growth.
• Existing numerical schemes often struggle with stability and accuracy, particularly for complex systems.
• Accurate and stable simulations are essential for understanding and predicting phenomena governed by phase-field equations.

Purpose of the Study:

• To develop unconditionally energy-stable, second-order time-accurate numerical schemes for diffuse-interface models.
• To apply these schemes to the Cahn-Hilliard equation and a diffuse-interface tumor-growth system.
• To demonstrate the accuracy, stability, and superiority of the proposed schemes compared to first-order methods.

Main Methods:

• Development of Crank-Nicolson type schemes incorporating a novel convex-concave splitting of free energy.
• Implementation of artificial-diffusivity stabilization for enhanced stability.
• Treatment of nonconstant mobility using extrapolation and semi-implicit handling of reactive terms for the tumor-growth model.

Main Results:

• The proposed schemes achieve unconditional energy stability and second-order time accuracy.
• Numerical examples confirm the theoretical accuracy and stability properties.
• The second-order schemes demonstrate superior performance compared to their first-order counterparts.

Conclusions:

• The developed numerical schemes provide a robust and accurate framework for simulating diffuse-interface phenomena.
• These methods are particularly effective for complex systems like tumor growth modeling.
• The unconditional energy stability and high-order accuracy pave the way for more reliable simulations in materials science and biology.