General propagation lattice Boltzmann model for nonlinear advection-diffusion equations

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Partial differential equations

Background:

• Nonlinear advection-diffusion equations (NADEs) are crucial for modeling various physical phenomena.
• Existing numerical methods face challenges with stability and accuracy, especially for variable coefficients and anisotropic diffusion.
• The standard lattice Bhatnagar-Gross-Krook (LBGK) model has limitations in handling complex NADEs.

Purpose of the Study:

• To propose a general propagation lattice Boltzmann model for nonlinear advection-diffusion equations (NADEs).
• To demonstrate the model's capability in recovering NADEs with variable coefficients.
• To enhance the stability and accuracy of numerical simulations for NADEs.

Main Methods:

• Development of a general propagation lattice Boltzmann model.
• Chapman-Enskog analysis to verify the recovery of NADEs with variable coefficients.
• Numerical simulations of linear advection-diffusion equation, nonlinear heat conduction equation, and NADEs with anisotropic and variable coefficients.

Main Results:

• The proposed model correctly recovers nonlinear advection-diffusion equations with variable coefficients, as confirmed by Chapman-Enskog analysis.
• Numerical simulations show excellent agreement between the model's results and analytical solutions for various advection-diffusion scenarios.
• The model demonstrates superior stability and accuracy compared to the standard lattice Bhatnagar-Gross-Krook model through parameter adjustment.

Conclusions:

• The general propagation lattice Boltzmann model provides a robust and accurate numerical tool for solving nonlinear advection-diffusion equations.
• Adjusting free parameters in the propagation step offers a pathway to optimize model performance for enhanced stability and precision.
• This model represents a significant advancement for simulating complex diffusion processes in scientific and engineering applications.