A Geometric Multigrid-Accelerated Compact Gas-Kinetic Scheme for Fast Convergence in High-Speed Flows on GPUs

Abstract

Implicit methods and GPU parallelization are two distinct yet powerful strategies for accelerating high-order CFD algorithms. However, few studies have successfully integrated both approaches within high-speed flow solvers. The core challenge lies in preserving the robustness of implicit algorithms in the presence of strong discontinuities, while simultaneously enabling massive thread parallelism under the constraints of limited GPU memory. To address this, we propose a GPU-optimized, geometric multigrid-accelerated, high-order compact gas kinetic scheme (CGKS) that incorporates three key innovations: (1) a multi-color lower-upper symmetric Gauss-Seidel scheme that eliminates thread conflicts and preserves memory efficiency, serving as an implicit smoother on coarse grids; (2) a discontinuity-adaptive relaxation technique and a multigrid prolongation process, based on a discontinuous feedback factor, which dynamically stabilize shock regions without compromising convergence in smooth zones; and (3) a three-layer V-cycle geometric parallel multigrid strategy specifically tailored for unstructured meshes. Extensive tests on multi-dimensional subsonic to hypersonic flows demonstrate that our GPU-based high-performance solver achieves one to two orders of magnitude faster convergence compared to previous explicit solvers. More importantly, it preserves the shock-capturing robustness of the explicit CGKS and exhibits strong scalability on GPU architectures. This work presents a unified framework that synergistically leverages implicit acceleration and GPU optimization for high-speed flow simulations, effectively overcoming traditional trade-offs between parallelism, memory constraints, and numerical stability in high-order methods.