[Paper Review] Comment on the paper Li-Shi Luo, Wei Liao, Xingwang Chen, Yan Peng and Wei Zhang, Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations, Physical Review E 83, 056710 (2011)

I. V. Karlin, Sauro Succi|arXiv (Cornell University)|Jul 15, 2011

Lattice Boltzmann Simulation Studies16 citations

TL;DR

This comment refutes a 2011 Physical Review E paper claiming the entropic lattice Boltzmann equation (ELBE) is inferior to standard LBGK models. The authors demonstrate that the simulation labeled as ELBE was actually a standard LBGK scheme with a constant relaxation time, not the true ELBE that uses an entropy-based, self-adaptive relaxation parameter. As a result, the comparison is invalid and circular, undermining the paper’s conclusions about ELBE’s stability and performance.

ABSTRACT

Critical comments on the entropic lattice Boltzmann equation (ELBE), by Li-Shi Luo, Wei Liao, Xingwang Chen, Yan Peng and Wei Zhang, Physical Review E 83, 056710 (2011), are based on simulations which make use of a model that, despite being called ELBE by the authors, is in fact fully equivalent to the standard lattice Bhatnagar-Gross-Krook equation. As a result, the conclusion of Luo et al on ELBE is circular, hence devoid of scientific bearing.

Motivation & Objective

• To correct the scientific misrepresentation in Luo et al. (2011), which claimed ELBE is less stable and inferior to LBGK.
• To clarify that the ELBE scheme is fundamentally different from LBGK due to its entropy-based, self-adaptive relaxation parameter governed by the H-theorem.
• To demonstrate that Luo et al. did not simulate true ELBE, but instead used a constant relaxation time, making their comparison invalid.
• To emphasize that ELBE’s true value lies in its ability to provide sub-grid stabilization via variable viscosity in under-resolved simulations, not in direct numerical simulations.
• To counter the claim that variable viscosity in ELBE is unphysical, arguing that such effective viscosity is common in computational fluid dynamics and essential for high-Reynolds-number flows.

Proposed method

• Identifying that Luo et al. used a constant relaxation time τ in their simulation, which violates the core mechanism of ELBE that relies on a dynamically determined relaxation parameter α via the entropy condition.
• Reiterating the correct ELBE formulation: populations evolve via f_i(x+v_i,t+1) - f_i(x,t) = αβ(f^eq_i - f_i), where α is determined by solving the entropy condition H(αf^eq + (1-α)f) = H(f), ensuring the H-theorem holds.
• Highlighting that the only difference between ELBE and LBGK lies in the equilibrium distribution and the relaxation parameter, with the latter being fixed in LBGK and adaptive in ELBE.
• Demonstrating that when populations are close to equilibrium (as in resolved simulations), ELBE reduces to LBGK with α = 2, making it identical to LBGK in the DNS regime.
• Arguing that the use of a polynomial approximation for the entropic equilibrium (as in Luo et al.) does not recover the true ELBE behavior, as the entropy condition (2) is not enforced.
• Citing independent implementations of ELBE by Keating et al., Spasov et al., Geerdink & Hoekstra, and Yasuda & Satofuka, which confirm the correct use of the entropy condition and self-adaptive relaxation.

Experimental results

Research questions

• RQ1 Does the entropic lattice Boltzmann equation (ELBE) with a constant relaxation time exhibit the same numerical stability as the standard LBGK model?
• RQ2 Is the ELBE scheme fundamentally different from the LBGK model when the relaxation parameter is fixed?
• RQ3 Can a simulation labeled as ELBE be scientifically valid if it does not enforce the entropy condition (2) that ensures the H-theorem?
• RQ4 Why is the claim that ELBE is 'the most inferior' among LB models scientifically unjustified?
• RQ5 Is the variable viscosity in ELBE unphysical, and does it compromise computational efficiency in a way that invalidates the method?

Key findings

• The simulation labeled as ELBE by Luo et al. (2011) is not ELBE, but a standard LBGK scheme with a constant relaxation time τ, rendering the comparison invalid.
• The authors of Luo et al. (2011) admit in their own paper that they did not test ELBE with a variable relaxation time, confirming their implementation was not the true ELBE.
• The entropy condition (2) is essential for ELBE’s H-theorem and sub-grid stabilization; its absence means the simulation cannot be considered ELBE.
• The difference between the entropic equilibrium and its polynomial approximation is of order O(u^4), which is within the truncation error of the lattice Boltzmann method for low Mach number flows.
• ELBE’s true strength lies in its self-adaptive stabilization in under-resolved simulations, where it deploys variable viscosity only when and where needed, unlike LBGK which fails under such conditions.
• The computational cost of solving the non-linear entropy condition is outweighed by the reduced grid resolution required at high Reynolds numbers, making ELBE more efficient overall in practical applications.