A class of lattice Boltzmann models for the Burgers equation with variable coefficient in space and time

Zongning Zhang; Chunguang Li; Jianqiang Dong; Zongning Zhang; Chunguang Li; Jianqiang Dong

doi:10.3934/math.2022251

AIMS Mathematics

2022, Volume 7, Issue 3: 4502-4516. doi: 10.3934/math.2022251

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Research article

A class of lattice Boltzmann models for the Burgers equation with variable coefficient in space and time

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
Zhengzhou University of Science and Technology, Zhengzhou, Henan 450000, China
3.
School of Civil Engineering, Hefei University of Technology, Hefei, Anhui 230009, China

Received: 11 October 2021 Revised: 30 November 2021 Accepted: 01 December 2021 Published: 22 December 2021
MSC : 35A07, 35A35

In this paper, we study the numerical results of the Burgers equation with the variable coefficient in space and time and then put forward a lattice Boltzmann model of backward difference solution of nonlinear system. The macroscopic equation is recovered by using the Chapman-Enskog method and the direct Taylor-series expansion method. These two methods can recover the same hydrodynamic equations and analyze various nonlinear systems. In particular, it is much easier to perform error analysis by using the direct Taylor method. In this study, the two methods are used to analyze the Burgers equation with variable coefficient in space and time, the numerical results are discussed and are compared with the analytical solution. The numerical results verify the effectiveness of the model. The stability of the model ensures that we can use larger time step lengths. The improvement of lattice speed can improve the computational performance of the model, and the D1Q7 lattice performance is much better than the D1Q5 lattice performance.
- lattice Boltzmann model,
- Chapman-Enskog method,
- direct Taylor expansion,
- variable coefficient
Citation: Zongning Zhang, Chunguang Li, Jianqiang Dong. A class of lattice Boltzmann models for the Burgers equation with variable coefficient in space and time[J]. AIMS Mathematics, 2022, 7(3): 4502-4516. doi: 10.3934/math.2022251

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Abstract

In this paper, we study the numerical results of the Burgers equation with the variable coefficient in space and time and then put forward a lattice Boltzmann model of backward difference solution of nonlinear system. The macroscopic equation is recovered by using the Chapman-Enskog method and the direct Taylor-series expansion method. These two methods can recover the same hydrodynamic equations and analyze various nonlinear systems. In particular, it is much easier to perform error analysis by using the direct Taylor method. In this study, the two methods are used to analyze the Burgers equation with variable coefficient in space and time, the numerical results are discussed and are compared with the analytical solution. The numerical results verify the effectiveness of the model. The stability of the model ensures that we can use larger time step lengths. The improvement of lattice speed can improve the computational performance of the model, and the D1Q7 lattice performance is much better than the D1Q5 lattice performance.

References

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