Fast and accurate coarsening simulation with an unconditionally stable time step

Area of Science:

• Computational physics
• Materials science
• Numerical analysis

Background:

• Phase-field modeling is crucial for simulating material microstructures.
• Existing numerical methods often require small time steps for stability and accuracy.
• Cahn-Hilliard and Allen-Cahn equations are fundamental in phase-field modeling.

Purpose of the Study:

• To develop unconditionally stable numerical integration algorithms for Cahn-Hilliard and Allen-Cahn equations.
• To achieve faster, accuracy-controlled simulations in phase-field modeling.
• To analyze the stability and accuracy of the proposed numerical methods.

Main Methods:

• Stability analysis using Eyre's theorem and von Neumann analysis.
• Development of accuracy-controlled integration with unbounded time steps (Δt ∝ t^α).
• Classification of time step exponents (α) and analysis of linear algorithms (α=1/3).

Main Results:

• The proposed algorithms are unconditionally stable, enabling faster simulations.
• Accuracy control is achieved with time steps that grow with time.
• A speedup of N/ln N is observed for a class of linear algorithms, reaching 300x faster than Euler for large systems.

Conclusions:

• The unconditionally stable algorithms provide significant speedups for phase-field simulations.
• The von Neumann stability analysis is straightforward and broadly applicable.
• These methods enhance the efficiency of simulating complex microstructural evolution.