Purified Two-Relaxation-Time Lattice Boltzmann Method: Removing Ghost Modes from TRT for Enhanced Stability

TL;DR

This work addresses the stability limitations of the two-relaxation-time lattice Boltzmann method by exposing ghost non-hydrodynamic modes as the root cause and establishing that ghost-mode filtering is equivalent to Hermite-based regularization within TRT-RLB. By leveraging an orthogonal decomposition of the velocity space into hydrodynamic and non-hydrodynamic subspaces, the authors prove exact equivalence for $D2Q9$ and $D3Q19$ lattices, enabling the Purified TRT (P-TRT) that subtracts ghost-mode contributions in a simple algebraic fashion. The approach preserves the accuracy and boundary properties of TRT-RLB while reducing computational cost (notably a 71% reduction in second-order non-equilibrium terms for $D2Q9$) and ensuring stability through ghost-eigenvalue annihilation ($z_G=0$). Comprehensive benchmarks—including double shear layers, Taylor–Green vortex decay, force-driven Poiseuille flow, and creeping flow past a square cylinder—confirm that P-TRT matches TRT-RLB in macroscopic behavior and accuracy while delivering substantial efficiency gains and robust stability at ultra-high Reynolds numbers. This ghost-mode viewpoint clarifies the core mechanism by which regularization stabilizes LBM schemes and suggests a path toward broader design of collision operators with guaranteed decoupling of non-hydrodynamic modes.

Abstract

The two-relaxation-time (TRT) lattice Boltzmann model is widely adopted for its simplicity and tunable boundary accuracy. However, its collision operator relaxes the full symmetric non-equilibrium component, implicitly retaining non-hydrodynamic ghost modes that degrade stability at high Reynolds numbers. In this work, we establish a rigorous connection between ghost-mode filtering and regularization within the TRT framework. By decomposing the discrete velocity space into hydrodynamic and non-hydrodynamic subspaces, we prove that the TRT-regularized lattice Boltzmann (TRT-RLB) model is mathematically equivalent to the standard TRT model with ghost modes explicitly removed. This equivalence holds exactly for D2Q9 and D3Q19 lattices, where the symmetric and antisymmetric subspaces are completely spanned by the physically relevant Hermite modes and identifiable ghost modes. Based on this finding, we propose the Purified TRT (P-TRT) model, which achieves regularization-level stability through simple algebraic ghost-mode subtraction rather than expensive tensor projections. For D2Q9, the non-equilibrium collision cost is reduced from 180 to 52 floating-point operations per node, a 71% reduction. Linear stability analysis in moment space further reveals that the P-TRT operator annihilates the ghost eigenvalue, proving its spectral radius is bounded above by that of standard TRT and that stability is governed exclusively by hydrodynamic modes. Benchmarks including the double shear layer at Re up to 10^7, Taylor--Green vortex decay, force-driven Poiseuille flow, and creeping flow past a square cylinder confirm that P-TRT preserves the stability, second-order accuracy, and zero-slip boundary properties of TRT-RLB while retaining the simplicity of the TRT family.

Figures (7)

• Figure 1: Maximum critical Mach number $Ma_c$ versus the inverse relaxation parameter $1/\tau_{s,2}$ in the double shear layer simulations at various Reynolds numbers. The P-TRT model (labeled "Present") demonstrates robust stability at super-high Reynolds numbers where standard BGK and TRT models fail to converge. The stability profile is identical to the original TRT-RLB model, confirming the mathematical equivalence.
• Figure 2: Comparison between the velocity components $u_x$ (left) and $u_y$ (right) computed by the P-TRT model and the analytical solution for the 2D decaying Taylor--Green vortex flow on a $64 \times 64$ grid with $\nu = 0.01$ at different times.
• Figure 3: Grid convergence study showing the $L_2$ error norms of the TRT-RLB and P-TRT models for the 2D decaying Taylor--Green vortex flow ($\nu = 0.01$) at $t = t_d$.
• Figure 4: The $L_2$ error versus the magic parameter $\Lambda_s$ for the P-TRT model. The vanishing error at $\Lambda_s = 3/16$ confirms the slip-elimination property inherited from the TRT framework.
• Figure 5: Comparison of horizontal velocity profiles between the P-TRT model and the analytical solution at $Re = 1, 5, 10$.

Theorems & Definitions (11)

• Remark 2.1
• Theorem 2.2: Second-order equivalence
• proof
• Theorem 2.3: Third-order equivalence
• proof
• Remark 2.4: Exactness for D2Q9 and D3Q19
• Remark 2.5: Approximation for D3Q27
• Theorem 2.6: Ghost-mode annihilation and stability decoupling
• proof
• Remark 2.7: Spectral radius comparison