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Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation
Xinmeng Chen1, Zhenhua Chai1,2, Jinlong Shang1
1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Physical Review. E
|October 16, 2021
Summary
A new multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) accurately solves nonlinear convection-diffusion equations. This stable and accurate method, along with its simplified version, offers improved performance over existing models.
Area of Science:
- Computational Fluid Dynamics
- Numerical Analysis
- Partial Differential Equations
Background:
- Nonlinear convection-diffusion equations (NCDEs) are crucial in modeling various physical phenomena.
- Existing numerical methods face challenges in accurately and stably solving NCDEs.
- The lattice Boltzmann method (LBM) offers a promising alternative for fluid dynamics and related problems.
Purpose of the Study:
- To develop a novel multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) for solving the nonlinear convection-diffusion equation (NCDE).
- To analyze the stability and accuracy of the proposed MRT-FDLBM and its variants.
- To compare the performance of MRT-FDLBM against existing numerical methods.
Main Methods:
- Development of a MRT-FDLBM by carefully designing the equilibrium distribution function and source term to ensure exact recovery of the NCDE.
- Application of von Neumann stability analysis to assess the stability of MRT-FDLBM and single-relaxation-time FDLBM (SRT-FDLBM).
- Introduction of a simplified MRT-FDLBM (SMRT-FDLBM) to reduce computational cost.
- Validation of the methods using a range of real and complex-value NCDEs, including Burgers-Fisher and Schrödinger equations.
Main Results:
- MRT-FDLBM and SMRT-FDLBM demonstrate second-order convergence rates in both spatial and temporal dimensions.
- Stability analysis reveals the order of stability from highest to lowest as: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, previous FDLBM, and LBM.
- Precision tests indicate the order of accuracy from highest to lowest as: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous FDLBM.
- SMRT-FDLBM achieves approximately a 15% reduction in computational cost compared to MRT-FDLBM.
Conclusions:
- The developed MRT-FDLBM provides an accurate and stable numerical solution for nonlinear convection-diffusion equations.
- The simplified SMRT-FDLBM offers a computationally efficient alternative without compromising accuracy.
- MRT-FDLBM and SMRT-FDLBM outperform existing finite-difference lattice Boltzmann methods in terms of stability and accuracy.
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