A time-accurate implicit simplified unified gas-kinetic scheme for unsteady multi-scale diatomic gas flows from continuum to free-molecular regimes

The unified gas-kinetic scheme (UGKS) solves multi-scale flows within a single framework, but its explicit formulation is inefficient for unsteady simulations due to severe Courant–Friedrichs–Lewy (CFL) constraints arising from disparate mesh scales. We develop a time-accurate implicit UGKS with a simplified multi-scale flux for unsteady diatomic gas flows. The method employs a Crank–Nicolson scheme with a dual-time-stepping strategy, enabling CFL numbers hundreds of times larger than those of the explicit scheme and achieving an order-of-magnitude speedup. The simplified multi-scale flux recovers the Navier–Stokes limit in the continuum regime and accurately resolves non-equilibrium effects under rarefied conditions. Systematic calibration of four key numerical parameters (CFL number, weighting factor ε, point-relaxation symmetric Gauss–Seidel sweeps, and inner iterations) using Sod's shock tube establishes practical guidelines for balancing accuracy and efficiency. The method is validated on benchmark problems spanning from continuum to free-molecular regimes: laminar flow past circular and square cylinders, flow expansion between two connected cavities, and nozzle plume flow. Numerical results demonstrate that the method accurately captures the distinct vortex-shedding dynamics of square and circular cylinders, thermal non-equilibrium under rarefied conditions, and flow patterns of the nozzle plume under different degrees of ambient rarefaction.