Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and...

Area of Science:

• Computational Materials Science
• Numerical Analysis
• Partial Differential Equations

Background:

• Materials science simulations often involve complex nonlinear higher-order parabolic partial differential equations.
• Efficient numerical methods are crucial for accurate and timely simulation of material behavior.

Purpose of the Study:

• To develop and evaluate efficient semi-implicit numerical methods for materials science models.
• To benchmark proposed methods against existing approaches for the Cahn-Hilliard equation.
• To explore higher-order time-stepping techniques for improved accuracy and convergence.

Main Methods:

• Implementation of semi-implicit methods for nonlinear higher-order parabolic partial differential equations.
• Convexity-splitting (CS) approach with backward Euler approximation.
• Comparison with the bi-harmonic-modified (BHM) approach.
• Introduction of higher-order time-stepping techniques.
• GPU implementation using MATLAB for computational acceleration.

Main Results:

• The proposed semi-implicit schemes demonstrate high efficiency for 2D computations.
• Demonstrated energy-decreasing property and overall performance for extended simulations in 2D and 3D.
• Achieved significant computational acceleration (up to 80x) for 3D Cahn-Hilliard equation simulations via GPU implementation.

Conclusions:

• The developed semi-implicit numerical methods are efficient and accurate for simulating materials science models like the Cahn-Hilliard equation.
• GPU acceleration offers a substantial speedup for complex 3D simulations.
• The methods provide a robust framework for long-term simulations with various initial conditions.