A macroscopic-shadow-corrected lattice Boltzmann method for fast, time-accurate simulation of low-Reynolds-number transient flows

Abstract

Explicit lattice Boltzmann simulations of slow transient flows are constrained by the acoustic time step. Dual-time stepping can remove this restriction, but slow inner convergence has limited reported wall-clock gains to roughly four- to tenfold, without a mechanism that strengthens under refinement. We present the macroscopic-shadow-corrected lattice Boltzmann method (MSC-LBM), which applies defect correction to the unsplit kinetic residual. At each Fourier wavenumber, a Stokes-like system is solved exactly for the conserved residual moments while preserving the second-order backward-differentiation formula (BDF2) fixed point and temporal accuracy. Along the two-relaxation-time axis $Λ=1/4$, $ω^+=ω^-=1$ makes one collision eliminate all non-hydrodynamic perturbations; $η(ν)=|6ν-1|/(6ν+1)$ vanishes at $ν=1/6$, leaving hydrodynamic slow modes whose long-wave shadow is inverted by the corrector. In completed timing campaigns, MSC-LBM achieves 27.95- and 52.71-fold wall-clock speedups at $N=256$, $\mathrm{Re}=1$ under matched-accuracy and $1\%$ gates. The matched speedup rises monotonically to 122.02 at $N=1024$, with gains persisting across $\mathrm{Re}=10^{-4}$--$100$. The three-dimensional D3Q19 extension reaches 27.85 and 8.84 under the same gates at $N=128$, $\mathrm{Re}=1$. A bounce-back-consistent kinetic coarse solver with adaptive relinearisation extends MSC-LBM to fully enclosed cavities, yielding setup-excluding physical-march speedups of 26.86 at $\mathrm{Re}=10$ and 3.84 at $\mathrm{Re}=100$ for $N=128$. An exact per-wavenumber symbol inverse establishes the attainable off-design contraction envelope. The contraction, parameter-sweep, and timing results jointly delimit the demonstrated operating regime: low-to-moderate-Reynolds-number transients for which acoustic stepping need not dictate computational cost.