General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows (Communications in Computational Physics, 2023)
In rarefied gas flows, the spatial grid size could vary by several orders ofmagnitude in a single flow configuration (e.g., inside the Knudsen layer it is at theorder of mean free path of gas molecules, while in the bulk region it is at a muchlarger hydrodynamic scale). Therefore, efficient implicit numerical method is urgentlyneeded for time-dependent problems. However, the integro-differential nature of gaskinetic equations poses a grand challenge, as the gain part of the collision operatoris non-invertible. Hence an iterative solver is required in each time step, which usually takes a lot of iterations in the (near) continuum flow regime where the Knudsennumber is small; worse still, the solution does not asymptotically preserve the fluiddynamic limit when the spatial cell size is not refined enough. Based on the generalsynthetic iteration scheme for steady-state solution of the Boltzmann equation, we propose two numerical schemes to push the multiscale simulation of unsteady rarefiedgas flows to a new boundary, that is, the numerical solution not only converges withindozens of iterations in each time step, but also asymptotically preserves the Navier-Stokes-Fourier limit in the continuum flow regime, when the spatial grid is coarse, andthe time step is large (e.g., in simulating the extreme slow decay of two-dimensionalTaylor vortex, the time step is even at the order of vortex decay time). The propertiesof fast convergence and asymptotic preserving of the proposed schemes are not onlyrigorously proven by the Fourier stability analysis for simplified gas kinetic models,but also demonstrated by several numerical examples for the gas kinetic models andthe Boltzmann equation.