Residual Carleman lift for nonlinear lattice Boltzmann simulations

Low-order Carleman embeddings replace nonlinear dynamics by a larger linear system, but truncation errors can accumulate even when the underlying nonlinear model is accurately approximated by a quadratic equation. We construct a residual Carleman lift (RC2) of lattice Boltzmann dynamics in which an independently evolved spectral background supplies the coefficients and forcing of a second-order residual lift. In a double-shear-layer test, conventional fixed-rest Carleman reaches $87.3\mathrm{\%}$ final error, whereas prepared RC2 achieves $0.136\mathrm{\%}$ through vortex roll-up, without restarting the lift or using fine-reference feedback. The results demonstrate control of the finite-time accuracy of a fixed-order linear embedding through nonlinear preparation.