# A Lattice Boltzmann BGK Model with an Amending Function for Two-Dimensional Second-Order Nonlinear Partial Differential Equations

## TL;DR

This paper introduces a new lattice Boltzmann model for solving complex nonlinear equations, showing it is efficient and accurate.

## Contribution

The novel contribution is a BGK lattice Boltzmann model with an amending function for second-order nonlinear PDEs.

## Key findings

- The model successfully recovers macroscopic equations via Chapman–Enskog analysis.
- Numerical experiments show excellent agreement with exact solutions.
- The model is robust for strongly nonlinear systems.

## Abstract

A mesoscopic lattice Boltzmann method based on the BGK model is proposed to solve a class of two-dimensional second-order nonlinear partial differential equations by incorporating an amending function. The model provides an efficient and stable framework for simulating initial value problems of second-order nonlinear partial differential equations and is adaptable to various nonlinear systems, including strongly nonlinear cases. The numerical characteristics and evolution patterns of these nonlinear equations are systematically investigated. A D2Q4 lattice model is employed, and the kinetic moment constraints for both local equilibrium and correction distribution functions are derived in the four velocity directions. Explicit analytical expressions for these distribution functions are presented. The model is verified to recover the target macroscopic equations in the continuous limit via Chapman–Enskog analysis. Numerical experiments using exact solutions are performed to assess the model’s accuracy and stability. The results show excellent agreement with exact solutions and demonstrate the model’s robustness in capturing nonlinear dynamics.