复杂可压缩流动中往往同时具有多尺度结构和间断结构，对数值格式和计算程序的设计都提出了苛刻的要求。对数值格式而言，一方面，多尺度结构要求数值格式具有良好的频谱特性（低耗散和低色散），以准确解析解的振幅和相位。另一方面，间断结构则要求数值格式具有足够的耗散，以捕捉间断结构而不产生数值振荡。对计算程序而言，实际流动的复杂性和工况的多样性要求计算程序不断拓展功能，以满足工程和学科发展的需求。本文针对复杂可压缩流动的高精度数值方法和计算程序开展了深入研究，主要研究内容和结论如下：

（1）提出了一种用于可压缩流动模拟的混合动理学通量WGVC（Weighted Group Velocity Control）-WENO（Weighted Essentially Non-Oscillatory）方法。该方法继承了文献\cite{liu2015}中提出的混合动理学方法，考虑了气体分子的自由传输和碰撞效应，并结合了文献\cite{he2014}中的WGVC-WENO重构技术。具有以下特点：在光滑区域，碰撞相关的通量起主要贡献，且具有较小耗散性，并采用WGVC重构方法；在非光滑区域附近，无碰撞的KFVS（Kinetic Flux Vector Splitting）通量起主要作用，具有较大耗散性，并与WENO重构技术相匹配。数值实验表明，该方法比文献\cite{liu2015}和其他FVS（Flux Vector Splitting）方法\cite{steger1981flux,harten1983upstream,kao2004lax}分辨率更高，耗散性更小。利用多个一维和二维算例，验证了该方法不仅具有高精度和低耗散特性，还展现出较强的鲁棒性和激波捕捉能力。

（2）为满足多尺度结构的分辨率要求，提出了一种优化的WGVC格式，该方法在保持精度的同时优化了数值格式的谱特性。通过设计波包的光滑度量因子和非线性加权方案，在低波数范围内实现精度控制，在中波数范围内实现群速度控制，最终显著提升了整体的谱特性。此外，通过将优化的WGVC格式嵌入到激波捕捉格式中，如WENO/TENO（Targeted Essentially Non-Oscillatory）格式，可以在增强格式激波捕捉能力的同时，保留了WGVC格式在中低波数的谱特性。理论和数值实验证明，优化的WGVC格式具有保持精度、色散和耗散误差小等优点，非常适合于湍流/激波边界层干扰等复杂流动问题的数值模拟。

（3）为实现动态非定常流动问题的高精度求解，基于OpenCFD-SC，开发了一款动态非定常高精度计算程序，程序利用刚性动网格和超限插值动网格实现了网格运动，利用局部时间步长方法和DP-LUR（Data-Parallel Lower-Upper Relaxation）方法实现高效的隐式并行计算。随后利用NACA0012翼型的俯仰振荡跨声速流动作为测试算例，对程序的正确性和收敛性进行了验证。计算结果与实验和他人数值模拟的结果基本一致，所开发的程序能够有效捕捉复杂的动态流动特征，为动态非定常流动问题的高精度数值模拟提供了研究基础。

（4）利用所开发的程序，开展了不同固定攻角和动态俯仰条件下的RAE2822超临界翼型的隐式大涡模拟研究。研究深入探讨了激波与湍流边界层的相互作用，揭示了随攻角变化激波强度和位置的运动规律，尤其在$5^\circ$攻角下，激波强度显著增强，且位置后移。动态俯仰运动中，激波呈现非标准的周期性运动，升力和力矩系数出现明显滞后。通过对比不同俯仰方向的流动特性和湍流特性，发现上仰过程加强了激波下游的流向速度脉动和正应力，削弱了其他方向的速度脉动及雷诺应力，而下俯过程则相反。通过对比不同攻角下的流动特性，发现动态流场特性受平均攻角影响很大，在合适的平均攻角和运动频率下，将有助于维持流动的稳定。

Complex compressible flows typically involve both multi-scale and discontinuous structures, posing stringent requirements on numerical schemes and computational algorithms. For numerical schemes, multi-scale structures require good spectral properties (low dissipation and dispersion) to accurately capture solution amplitudes and phases. In contrast, discontinuous structures demand sufficient dissipation to resolve discontinuities while avoiding numerical oscillations. For the computational algorithms, the complexity of actual flows and the diversity of operating conditions require continuous expansion of functionality to meet the needs of engineering and academic development. This study focuses on high-order numerical methods and computational algorithms for compressible flows, with the main research content and conclusions as follows:

(1) A high-order hybrid kinetic flux WGVC (Weighted Group Velocity Control) - WENO (Weighted Essentially Non-Oscillatory) method for compressible flow simulations is proposed. This method is based on the hybrid kinetic method proposed by Liu\cite{liu2015}, which considers both free transport and collision effects of gas molecules. It further integrates the WGVC-WENO reconstruction technique from He et al.\cite{he2014}. The method operates as follows: In smooth regions, the flux related to collisions plays the main role, with low dissipation, using WGVC reconstruction; near discontinuities, the collisionless KFVS (Kinetic Flux Vector Splitting) flux dominates, with higher dissipation, matching the WENO reconstruction technique. Numerical experiments indicate that this method achieves higher resolution and lower dissipation compared to \cite{liu2015} and other FVS (Flux Vector Splitting) methods. Using multiple one- and two-dimensional cases, the method demonstrates not only high order and low dissipation but also strong robustness and shock-capturing capabilities.

(2) To meet the resolution requirements of multi-scale structures, an optimized WGVC scheme is proposed. This method improves the spectral properties of the numerical scheme while maintaining order. By utilizing smoothness indicators and a nonlinear weighting approach for wave packets, it enables order control in the low-wavenumber range and group velocity control in the mid-wavenumber range. Additionally, embedding the optimized WGVC scheme within shock-capturing schemes, such as WENO/TENO (Targeted Essentially Non-Oscillatory) schemes, the spectral properties of the WGVC scheme at mid-to-low-wavenumbers are preserved while enhancing the shock-capturing capability of the scheme. Theoretical analysis and numerical experiments confirm that the new method offers benefits such as order preservation, minimal dispersion and dissipation errors, and is highly suitable for simulating complex flow problems like turbulence and shock-boundary-layer interactions.

(3) To achieve high-order solutions for dynamic unsteady flow problems, a dynamic high-order computational program was developed based on OpenCFD-SC. The program implements grid motion using rigid body moving grids and transfinite interpolation grids, and achieves efficient implicit parallelization through local time-stepping and the DP-LUR (Data-Parallel Lower-Upper Relaxation) method. Using the pitching oscillation of a NACA0012 airfoil in transonic flow as a benchmark case, the program’s correctness and convergence were validated. The simulation results align well with experimental data and other numerical results, confirming that the developed program can effectively capture complex dynamic flow features, providing strong support for high-order numerical simulations of complex unsteady flow problems.

(4) The developed program was used to conduct ILES (Implicit Large Eddy Simulation) studies on the RAE2822 supercritical airfoil under various fixed angles of attack and dynamic pitching conditions. The study thoroughly investigated the interaction between shock waves and turbulent boundary layers, focusing on how shock intensity and position vary with changing angles of attack. At a $5^\circ$ angle of attack, the shock intensity increased significantly, and its position shifted downstream. In dynamic pitching motion, the shock exhibited a non-standard periodic motion, and the lift and moment coefficients showed significant hysteresis. By comparing the flow and turbulence characteristics in different pitching directions, it was found that the nose-up process enhanced the streamwise velocity fluctuations and normal stress downstream of the shock, while suppressing fluctuations and Reynolds stress in other directions; the nose-down process had the opposite effect. Analyzing the flow characteristics at different angles of attack showed that the dynamic flow characteristics were strongly influenced by the mean angle of attack, and flow stability could be promoted at an appropriate mean angle and pitching frequency.