Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations:...

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Mathematical physics

Background:

• The multiple-relaxation-time lattice Boltzmann (MRT-LB) method is a powerful numerical technique for simulating fluid flow and diffusion phenomena.
• Existing analysis methods for deriving macroscopic equations from lattice Boltzmann models can be complex and varied.

Purpose of the Study:

• To present a unified framework for the multiple-relaxation-time lattice Boltzmann (MRT-LB) method applicable to Navier-Stokes and nonlinear convection-diffusion equations.
• To compare four popular analysis methods for deriving macroscopic equations from the MRT-LB method.
• To provide implementation details for the MRT-LB method and identify potential existing lattice Boltzmann models.

Main Methods:

• Development of a unified MRT-LB framework incorporating a block-lower-triangular-relaxation matrix and an auxiliary source distribution function.
• Comparative analysis of Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches.
• Mathematical derivation and verification of macroscopic equations.

Main Results:

• The unified MRT-LB framework successfully accommodates both Navier-Stokes and nonlinear convection-diffusion equations.
• All four compared analysis methods yield identical macroscopic equations at the second-order of expansion parameters from a mathematical perspective.
• Key implementation elements for the MRT-LB method are identified, and connections to existing lattice Boltzmann models are established.

Conclusions:

• The proposed unified MRT-LB framework offers a generalized approach for simulating complex fluid dynamics and diffusion processes.
• The equivalence of different analysis methods simplifies the theoretical understanding and application of MRT-LB models.
• The findings facilitate the practical implementation and further development of MRT-LB methods.