A Compact Gas-Kinetic Scheme with Scalable Geometric Multigrid Acceleration for Steady-State Computation on 3D Unstructured Meshes

Abstract: In this paper, we present an advanced high-order compact gas-kinetic scheme (CGKS) for 3D unstructured mixed-element meshes, augmented with a geometric multigrid technique to accelerate steady-state convergence. The scheme evolves cell-averaged flow variables and their gradients on the original mesh. Mesh coarsening employs a two-step parallel agglomeration algorithm using a random hash for cell interface selection and a geometric skewness metric for deletion confirmation, ensuring both efficiency and robustness. For the coarser meshes, first-order kinetic flux vector splitting (KFVS) schemes with explicit or implicit time-stepping are used. The proposed multigrid CGKS is tested across various flow regimes on hybrid unstructured meshes, demonstrating significant improvements. A three-layer V-cycle multigrid strategy, coupled with an explicit forward Euler method on coarser levels, results in a convergence rate up to ten times faster than standard CGKS. In contrast, the implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method offers limited convergence acceleration. Our findings indicate that the explicit multigrid CGKS is highly scalable and effective for large-scale computations, marking a substantial step forward in computational fluid dynamics.