General propagation lattice Boltzmann model for the variable-coefficient Gardner equation

This study presents a general propagation lattice Boltzmann model for solving the variable-coefficient Gardner equation, a nonlinear evolution equation describing weakly nonlinear long-wave propagation in KdV-type media. The proposed model integrates the Lax–Wendroff technique and fractional propagation methodology within a time-splitting framework, enhancing computational stability through adjustable parameters. By systematically deriving the equilibrium distribution function, compensation terms, and source components, the model accurately recovers the macroscopic equation via the Chapman-Enskog analysis. Numerical simulations validate the model’s accuracy and stability, with optimization methods (the Nelder–Mead algorithm and the Broyden–Fletcher–Goldfarb–Shanno method) employed to determine optimal free parameters. The results demonstrate excellent agreement with analytical solutions, highlighting the model’s potential for handling complex nonlinear systems with variable coefficients.