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Updated: Jul 21, 2026
05:20
Characterization of Thermal Transport in One-dimensional Solid Materials
Published on: January 26, 2014
From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models
Paulo C Philippi1, Luiz A Hegele, Luís O E Dos Santos
1LMPT Mechanical Engineering Department, Federal University of Santa Catarina, Florianópolis, Brazil. philippi@lmpt.ufsc.br
Summary
This study presents a novel approach to velocity discretization for the lattice Boltzmann method (LBE), ensuring minimal discrete velocity sets for accurate approximations of the continuous Boltzmann equation. The findings reveal new space-filling lattices with enhanced dimensionality for thermal LBE simulations.
Area of Science:
- Computational Fluid Dynamics
- Statistical Mechanics
- Numerical Analysis
Background:
- The continuous Boltzmann equation is fundamental to fluid dynamics.
- Velocity discretization is a crucial step in deriving the lattice Boltzmann method (LBE).
- Existing discretization methods may not always yield optimal discrete velocity sets.
Purpose of the Study:
- To develop an alternative approach for velocity discretization in LBE.
- To determine minimal discrete velocity sets based on approximation order and lattice structure.
- To investigate the relationship between discrete and continuous inner products for Hermite polynomial tensors.
Main Methods:
- Analysis of velocity discretization within the framework of the continuous Boltzmann equation.
- Equivalence established between discretization and discrete inner product realization.
- Preservation of norm and orthogonality of Hermite polynomial tensors in Hilbert space.
Main Results:
- Demonstrated that even-parity velocity tensors are isotropic up to rank 2N for approximation order N.
- Derived space-filling lattices with increased dimensionality compared to existing ones.
- Presented and discussed two-dimensional square lattices for thermal LBE problems.
Conclusions:
- The proposed method provides a systematic way to obtain optimal discrete velocity sets.
- The resulting lattices offer advantages in terms of dimensionality and accuracy for LBE simulations.
- This work contributes to the advancement of numerical methods for Boltzmann equation-based simulations.
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