高精度、高分辨率计算方法一直是计算流体力学中的重点研究问题。超声速流动具有复杂的流动特征，其数值模拟不仅需要有效的捕捉流场中的激波结构，同时还需要高精度、高分辨率的模拟出各种激波干扰、分离流动、漩涡生成、脱落的演化等对飞行器气动性能有重要影响的复杂流动现象。捕捉激波需要引入数值耗散，而在无激波的光滑区域过大的数值耗散常会掩盖真实的物理耗散，影响数值模拟的计算精度。因此，发展超声速复杂流动的数值模拟方法的一个关键且具有挑战性的难题，就是怎样在保持无振荡的捕捉激波间断的同时能高精度、低耗散的模拟无激波的光滑区域中的各种流动现象。

加权基本无振荡（weighted essentially non-oscillatory简称WENO）格式具有较高的计算精度，而且能无振荡的捕捉激波间断，近年来已成为模拟研究超声速流动的主要数值方法。本文研究主要以WENO格式为基础，针对其在应用中的不足，发展性能更好的相应方法。

1) 发展了完全五阶的低耗散WENO-Z型格式。五阶WENO-Z格式为发展高阶格式提供了一个直接的分析方法，但是其在保持光滑区（包括极值点）的一致高精度和间断附近的低耗散两方面存在矛盾，无法兼顾。本文提出了一种光滑度自调节函数，使得在光滑区域所计算的权值能以更高阶的精度逼近最优权，在光滑区域（包括极值点）满足五阶收敛的充分条件，在间断区域保持WENO-Z的低耗散及无振荡性质。该光滑度自调节函数可推广到高阶WENO格式的构造，为发展高精度、低耗散WENO格式提供了一种新颖有效的方法。

2) 提出了无人为参数的高精度间断模板判别方法，并发展了有限紧致-WENO格式。有限紧致-WENO格式同时具有紧致格式的高精度、低耗散和WENO格式的高精度激波捕捉两个优点，但是其计算效果严重依赖于其使用的间断判别方法。对此，本文通过构造性能更好、精度更高的五点模板的整体光滑因子，并利用其与子模板光滑因子之间的量级差发展了不含人为参数的五点间断模板判别方法，该方法能有效分辨间断模板与光滑模板（包含一阶极值点模板及二阶极值点），由此发展的有限紧致-WENO格式具有很高的计算精度和鲁棒性，为模拟激波/复杂流动干扰提供了一个有效的高精度、低耗散方法。

3) 针对三阶WENO格式，提出了利用整体模板的三个模板点来计算各子模板光滑因子的方法，突破了现有三阶加权格式仅利用子模板上的两个结点来计算子模板光滑因子的局限性，所构造的两个新子模板光滑因子在单调光滑区域完全相等，因此加权过程不会引入额外的耗散，极大地降低了加权格式的数值耗散。

4) 首次揭示了三阶WENO格式精度低、耗散大的本质原因是其子模板光滑因子不能正确对待含隐式极值点的模板（隐式极值点是指极值点不是计算模板上的网格结点）。在此基础上，本文提出了无人为参数的四点模板间断判别方法并结合三点间断判别方法来识别含隐式极值点的模板与真正的三点间断模板，发展了三阶WENO混合格式。分析及数值算例表明，本文发展的判别方法能有效辨别含隐式极值点模板与间断模板，所得WNEO混合方法在光滑区域（包括极值点）获得了完全一致的三阶精度。由于三阶WENO格式相比更高阶的WENO格式具有鲁棒性好、所用模板点少、边界处理简单等优点，因此本文发展的真正具有三阶精度的WENO混合格式为工程实际应用提供了一个有效的数值方法。

5) 在模拟超声速流动时，低耗散的近似黎曼求解方法，如Roe分裂方法，常会产生数值激波不稳定问题，得到非物理解。为了抑制这类数值激波紊乱，本文以发展的四点间断模板判别方法为基础，构造了无人为参数的激波判别方法，并发展了混合Roe-LF分裂方法：即在激波界面使用LF分裂抑制数值激波不稳定现象，而在其他区域使用Roe分裂方法维持其高分辨率的接触间断和剪切流动捕捉效果。数值算例表明，本文发展的激波判别方法和Roe-LF分裂具有很好的鲁棒性，不仅具有Roe分裂的低耗散特性，还能有效抑制各种数值激波不稳定问题。

6) 初步将本文发展的高精度高分辨率算法应用于求解三维跨声速和超声速流动问题。计算结果表明，本文发展的算法不仅具有较高的计算精度，还具有较强的激波捕捉能力，能清晰模拟流场中的激波、激波/边界层干扰以及流动分离、再附等复杂流动现象。

Developing the high order high resolution numerical methods is aways an important issue in computational fluid dynamics. Supersonic flows involve complex flow characteristics, such as the interactions between shock waves, flow separation, the formation,shedding and evolution of vortex. The methods for supersonic flows are required not only to capture shock with essentially non-oscillation, but also to resolve the complex multiscale structures with high order accuracy and high resolution. Generally, it is necessary to introduce numerical dissipation for capturing shock waves, while excessive numerical dissipation would overwhelm the real physical dissipation in smooth regions without shock waves and hence decrease the accuracy. Therefore, how to design a method which can maintain non-oscillatory properties around shock regions and achieve high accuracy and high resolution in smooth regions simultaneously is particularly challenging.

The weighted essentially non-oscillatory (WENO) schemes can obtain high order accuracy in the smooth regions and keep ENO property near shock waves. Hence, the WENO schemes have been become the main numerical method for simulating the complex flows. This paper mainly focuses on developing high performance WENO schemes and their applications in complex flows.

1) This paper proposed a new low dissipation full-fifth order WENO scheme. The fifth order WENO-Z scheme provides a straightforward way to analyze the performance of high order WENO schemes, however, it cannot achieve both the high order accuracy at critical points and low dissipation property near discontinuities. To solve this dilemma, this paper proposed a stencil smoothness adaptive function, which can improve the accuracy order of the calculated weights approximating to the optimal weights. The resulting scheme satisfies the sufficient condition for fifth order convergence in smooth solutions (including critial points), and also maintains the low dissipation and ENO property of the WENO-Z scheme in discontinuous regions. The stencil smoothness adaptive function can be easily extend to higher-order schemes, hence it provides a novel and effective strategy for developing high accuracy high resolution WENO schemes.

2) A new detecting method for discontinuous stencil is proposed and used to construct a new finite compact WENO (FC-WENO) scheme. The FC-WENO scheme can achieve both the high order high resolution of a compact scheme and the shock capturing ability of a WENO scheme, while, its behavior depends on the shock detector. Therefore, this paper developed a high order global stencil smoothness indicator and used to design a parameter-free five-points stencil discontinuity detecting method combining with the local smoothness indicators. The new detector can distinguish well the discontinuous stencils and smooth stencils (including those containing first-order and second-order critical points), and the resulting FC-WENO scheme is quite efficient, accurate and robust for complex shock interaction flow.

3) This paper proposed a novel way for the third order WENO scheme to calculate the local smootheness incidators (LSIs) by using the three points on the global stencil. This method breaks through the limit of traditional weighting method, which only uses the two points on each local stencil to calculate the LSIs. The resulting two new LSIs are equivalent in monotonic smooth regions, hence the weighting process does not introduce extra dissipation, and hence greatly decrease the numerical dissipation.

4) This paper first reveals the inherent reason for the low accuracy and large dissipation of the third order WENO scheme: their local smoothness indicators cannot correctly treat the smooth three-point stencil containing an implicit critical point (implicit means the critical point is not a grid point). To overcome this shortcoming, this paper designed a four-points stencil discontinuity detecting method and applied to construct a third-order hybrid WENO scheme. The analyses and numerical results show that, the new detecting method can effectively distinguish discontinuous stencils and the smooth ones containing implicit points, and the hybrid scheme can gain full third order accuray including critical points. Compared with higher order WENO schemes, the third order one is more robust for shock problems, uses less grid points, and reduces the difficulty of the boundary treatment. Hence, it is more suitable for complicated engineering problems.

5) The numerical shock insstability always occurs in the simulation of supersonic flows by using low dissipation Riemann solvers(for example, the Roe scheme). To suppress the spurious phenomenon, this paper designed a shock detector based on the four-points discontinuity detecting method and used to develop the hybrid Roe-LF flux splitting method, in which, the LF is used in shock regions to remove the shock instability and the Roe is used in smooth regions to preserve the high resolution around contact discontinuities and shear flow. The numerical results show that, the new shock detector and the Roe-LF method are robust, accurate and efficient for suppressing the numerical shock instability.

6) The high order high resolution schemes developed in this paper are applied to solve the three-dimensional complex flows, including the transonic and supersonic flows. The results show that, the schemes developed in this paper not only have high order accuracy, but also can capture strong shocks without oscillatory, they can correctly simulate the complex flow structure, such as the shock, shock/bounary interaction and seperation, reattachment flows.