A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients.

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Partial differential equations

Background:

• Fourth-order partial differential equations (PDEs) with variable coefficients model complex phenomena.
• Existing numerical methods may struggle with stability and accuracy for such PDEs.

Purpose of the Study:

• To develop a unified lattice Boltzmann model for fourth-order PDEs with time-dependent variable coefficients.
• To ensure the model's accuracy and stability for diverse PDE problems.

Main Methods:

• A unified lattice Boltzmann model was formulated.
• A compensation function was integrated into the evolution equation.
• Chapman-Enskog and Taylor expansion methods were applied.
• Error analysis was conducted to determine accuracy.

Main Results:

• The macroscopic equation was correctly recovered.
• The proposed model demonstrated second-order accuracy in time.
• Simulations of constant- and variable-coefficient PDEs were successful.

Conclusions:

• The unified lattice Boltzmann model is effective and stable.
• The model provides a reliable numerical approach for solving complex fourth-order PDEs.
• This work advances numerical techniques for fluid dynamics and related fields.