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.

翻译：在先前的研究中，高阶气体动理学格式（HGKS）已在三维非结构网格的非稳态流动中取得成效。本文为加速稳态流动的收敛速度，发展了隐式非紧致和紧致HGKS。对于非紧致格式，采用简单加权本质无振荡（WENO）重构实现空间精度，其重构模板包含两层相邻单元。结合非线性广义最小残差（GMRES）方法，构建了隐式非紧致HGKS。为提升非紧致格式的分辨率与并行性，发展基于Hermite WENO（HWENO）重构的隐式紧致HGKS，其重构模板仅含一层相邻单元。单元平均守恒变量同样通过GMRES方法更新，同时采用简单策略利用时空耦合气体分布函数的时间演化更新单元平均梯度。为加速计算，基于统一计算设备架构（CUDA）在图形处理器（GPU）上实现了隐式非紧致和紧致HGKS。通过从亚声速到超声速流动的多种数值算例，验证了格式对无黏与黏性流动的精度、鲁棒性和计算效率。