An asymptotic-preserving adjoint unified gas kinetic scheme for sensitivity analysis

Abstract

High-dimensional sensitivity analysis and uncertainty quantification for multiscale gas dynamics, spanning the continuum to rarefied regimes, require computationally efficient and mathematically consistent gradient evaluation. This paper develops a discrete adjoint method for the unified gas-kinetic scheme (UGKS) based on a dual-consistent formulation. The adjoint system is derived directly from the discrete microscopic velocity-distribution equation coupled with the macroscopic-moment compatibility conditions. To resolve the stiff cross-scale coupling, we propose an asymptotic-preserving (AP) adjoint formulation constructed via macroscopic-moment projection and microscopic lifting. Under this framework, the AP adjoint formulation eliminates the stiff collision coupling and removes the collision-time step restriction in the continuum regime. Numerically, a memory-efficient residual-evaluation algorithm that mirrors the forward UGKS cell-vertex data structure is implemented to bypass the memory bottleneck in velocity space. Furthermore, a macroscopic--microscopic predictor--corrector implicit marching scheme is designed to accelerate convergence without solving a globally coupled system. The accuracy, consistency, and robustness of the proposed AP-adjoint scheme are rigorously verified against an independent linearized UGKS solver across a wide range of Knudsen numbers, including lid-driven cavity heat conduction, microchannel thermal creep flow, and hypersonic flow past a circular cylinder.