Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes I: steady flows

TL;DR

This work tackles slow convergence in steady compressible flow simulations on three-dimensional unstructured meshes by developing two implicit gas-kinetic schemes (HGKS): a non-compact WENO-based method and a compact HWENO-based method. Each scheme uses a GMRES-based time-stepping approach with Roe-type Jacobians and Jacobi preconditioning, and updates cell-averaged variables (and gradients for the compact scheme) through the gas-kinetic fluxes derived from the BGK model, improving convergence for steady states. The methods are GPU-accelerated via CUDA, achieving significant speedups while maintaining high-order accuracy across subsonic to supersonic regimes, including flows around spheres and the ONERA M6 wing. The results demonstrate robust performance and suggest the approach is well-suited for large-scale, high-Reynolds-number simulations on complex geometries, with potential for multi-GPU extensions. Overall, the paper delivers a pair of high-order, implicit HGKS solvers that combine accuracy, stability, and parallel efficiency for steady compressible flows on unstructured meshes.

Abstract

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows.

Figures (16)

• Figure 1: The non-zero block distribution of $\boldsymbol{A}$.
• Figure 2: Connection between threads and unstructured cells.
• Figure 3: Lid-driven cavity flow: the local computational mesh distribution with tetrahedral meshes.
• Figure 4: Lid-driven cavity flow: the steady state $U$-velocity profiles along the vertical centerline (left), $V$-velocity profiles along the horizontal centerline (right) for $Re=1000$ and $100$.
• Figure 5: Lid-driven cavity flow: the residual comparison with LUSGS-WENO, GMRES-WENO and GMRES-HWENO methods for $Re=100$ and $1000$.