# Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations

## TL;DR

This paper introduces a discrete unified gas kinetic scheme (DUGKS) for nonlinear convection-diffusion equations, demonstrating its accuracy, stability, and flexibility with non-uniform meshes through analytical and numerical comparisons.

## Contribution

The paper develops a novel DUGKS for NCDEs, showing its second-order convergence, compatibility with non-uniform meshes, and superior stability and accuracy over existing lattice Boltzmann models.

## Key findings

- DUGKS achieves second-order convergence.
- DUGKS is more accurate than FV-LBM.
- DUGKS is more stable than SLBM.

## Abstract

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for general nonlinear convection-diffusion equation (NCDE), and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and find that the numerical results are in good agreement with analytical solutions and the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, DUGKS can also adopt the non-uniform mesh. Besides, we performed some comparisons among the DUGKS, finite-volume lattice Boltzmann model (FV-LBM), single-relaxation-time lattice Boltzmann model (SLBM) and multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the DUGKS model is more accurate than FV-LBM, more stable than SLBM, and almost has the same accuracy as the MRT-LBM. Besides, the using of non-uniform mesh may make DUGKS model more flexible.