Rigorously justified local time stepping in the unified gas-kinetic wave-particle method for steady multiscale flow simulation

Abstract

Local time stepping (LTS) can accelerate convergence to steady states in kinetic simulations with large variations in the local time steps across the computational domain. When neighboring cells advance with unequal time steps, the time-averaged particle flux must be balanced across their common interface. For particle-based or hybrid wave-particle methods under finite volume method (FVM) framework, we rigorously establish a sufficient condition for time-averaged interfacial particle-flux balance: fixed positive local time steps together with proportional particle-mass scaling. When a particle crosses from cell $L$ to cell $R$, its mass is scaled by $\Delta t_R/\Delta t_L$. In the unified gas-kinetic wave-particle (UGKWP) implementation, the same ratio is applied to the remaining free-transport time of the crossing particle. LTS also affects the wave-particle decomposition and the time integration of the wave fluxes in UGKWP. The wave-particle decomposition in cell \(i\) is determined by the local ratio \(\Delta t_i/\tau_i\), which better reflects the local relation between the observation scale and relaxation time for multiscale cases. The equilibrium and analytic free transport wave fluxes are integrated over and normalized by the corresponding cell-side time steps to obtain the interfacial time-averaged wave fluxes. The UGKWP-LTS method is used to simulate the hypersonic flow past a cylinder at $\mathrm{Kn}=0.01$ and $0.1$, and a flat plate at $\mathrm{Kn}=0.0169$. In all three cases, the surface quantities obtained with UGKWP-LTS agree well with the reference data. Relative to global time stepping (GTS), UGKWP-LTS achieves step-count speedups of $6.6\times$, $3.8\times$, and $20\times$ for the three cases, respectively. The corresponding wall-clock speedups are $7.1\times$, $4.5\times$, and approximately $21\times$.