超声速流动具有复杂的流动特征，如激波、激波/复杂流动相互作用、粘性/无粘干扰效应、流动转捩等。其中，为了无振荡捕捉激波，所使用的数值方法通常要引入数值耗散，而为了分辨流场中的其它小尺度结构，又要求方法具有高精度、低耗散等特性，计算流体力学中高精度高分辨率激波捕捉格式的研究也正是围绕解决这一矛盾而得到了不断的发展。加权基本无振荡（Weighted essentially nonoscillatory，简称 WENO）格式是近年来高精度激波捕捉格式的突出代表，自提出以来得到了不断的发展和广泛的应用。本文进一步发展了性质更好的WENO格式，如摄动 WENO 格式，多步 WENO 格式等，并应用于超声速平板边界层的转捩预测等研究。
1) 提出和发展了摄动加权基本无振荡格式。为了提高 WENO 格式的精度及耗散色散特性，传统方法基本都是通过改进模板光滑因子或发展新的权值计算来实现，而摄动 WENO 格式则通过数值摄动子模板的数值通量函数，以提高子通量函数的精度，再将摄动后的数值通量进行加权，获得最终的摄动加权格式。首先分析了双曲守恒方程守恒型格式中数值通量的精度，并给出了一个关于数值通量精度的引理，在此基础上利用数值摄动思想，发展了高阶的子模板数值通量函数；接着，对摄动数值通量通过非线性权的加权凸组合方式构造了五阶摄动WENO（Perturbational WENO，简称 P-WENO）格式的最终数值通量。理论分析表明，五阶 P-WENO 格式降低了传统五阶 WENO 格式五阶收敛的充分必要条件和充分条件对权值的要求，即 P-WENO 格式更容易达到五阶收敛，有效解决了五阶 WENO 格式极值点五阶精度和间断附近低耗散特性无法同时满足的问题。最后，我们推广并发展了七阶 P-WENO 格式。摄动加权方法为发展 WENO 格式提供了一个新的构造思路。
2) 发展了高性能五阶多步 WENO 格式。针对传统五阶 WENO 格式在过渡点（连接光滑区域和间断点的点）精度降阶问题，本文分析了五阶 WENO 格式在过渡点的精度，在对已有的相关改进方法综合对比分析的基础上重新构造了新的权值计算方法，并以此发展了高性能五阶多步 WENO 格式。理论分析表明，新
格式在光滑区（包括极值点）满足五阶收敛的充分必要条件，同时将过渡点精度
提高了一阶。新格式形式简单，既提高了原多步 WENO 格式的计算效率，同时也避免了其它多步 WENO 格式在间断附近计算精度降低的问题。
3) 发展了低色散、低耗散三阶 WENO 格式。与高阶 WENO 格式相比，三阶WENO 格式具有鲁棒性好、边界易于处理等优点。然而，现有三阶 WENO 格式仍然存在数值耗散较大等问题。为进一步发展性能更优的三阶 WENO 格式，本文分析了不同三阶 WENO 格式中权值与子模板光滑因子之间的函数关系，在此基础上构造了新的函数，并用该函数发展了低色散、低耗散三阶 WENO 格式。新格式极大地提高了非线性权值逼近理想权值的精度，且谱性质分析和数值试验
结果均表明其具有低色散和低耗散特性。
4) 探索了抛物化 NS（Parabolized Navier Stokes，PNS）方程应用于超声速平板边界层转捩问题的直接数值模拟。抛物化 NS 方程不仅能反应边界层的特征，同时也考虑了粘性与无粘的相互干扰，另一方面，由于抛物化 NS 方程只保留除近似主流方向之外的粘性项，因而也为数值离散带来极大的方便。考虑到流动转捩通常发生在边界层内，因此研究抛物化 NS 能否正确模拟超声速边界层的转捩问题无疑具有重要意义。本文针对超声速平板边界层转捩模拟中的两种条件，即添加吹吸扰动和设置粗糙元，利用本文发展的高精度格式对 PNS 方程进行了直接数值模拟，并与完全 NS 方程的数值结果进行了对比，结果表明 PNS 方程可以模拟超声速平板边界层的转捩现象。

Supersonic flow has complex flow characteristics, such as shock waves, shock waves/complex flow interactions, viscous/inviscid interactions, flow transitions. In order to capture the shock wave without oscillation, the numerical dissipation is usually introduced into the numerical methods. Meanwhile, in order to resolve other smallscale structures in the flow field, the method is required to have high accuracy and low dissipation. The high accuracy and high resolution shock-capturing schemes in computational fluid dynamics are developing all the time for dealing with this contradiction.
    The weighted essentially non-oscillatory (WENO) scheme is a prominent representative of the high accuracy shock-capturing schemes in recent years, which has been continuously developed and widely used in CFD since it was proposed. In this paper, we further develop the WENO scheme with novel idea and advanced properties, such as perturbational WENO scheme, multistep WENO scheme. In addition, these methods are also applied to simulate the transition of supersonic flat-plate boundary layer in this work.
1) A perturbational weighted essentially non-oscillatory scheme is proposed and developed. In order to improve the accuracy and dissipative dispersion characteristics of the WENO scheme, the traditional methods are basically realized by improving the smoothness indicator of the sub-stencil or developing new method for calculating weights.
However, the perturbational WENO scheme improves the accuracy of the sub-flux function by using the numerical perturbation sub-stencils numerical flux function, and then the perturbation weighted scheme is constructed by weighting the perturbed numerical fluxes. Firstly, the accuracy of the numerical flux in the conservative form of the hyperbolic conservation equation is analyzed, and a lemma about the accuracy of numerical flux is given. The high order sub-stencil numerical fluxes are obtained by using the idea of numerical perturbation algorithm. And then, the final numerical flux of a fifth order perturbational WENO (P-WENO) scheme is constructed by a convex combination of the new perturbed fluxes. Theoretical analysis shows that the P-WENO scheme relaxes
the requirement of necessary and sufficient conditions for fifth-order convergence on the weights. That is, the P-WENO scheme is easier to obtain the fifth-order convergence, which effectively solves the problem that the fifth-order WENO scheme is difficult to obtain low dissipation near discontinuity while achieving fifth-order accuracy at critical point. Finally, the new method is generalized and a seventh-order P-WENO scheme isdeveloped. The perturbation weighting method provides a novel idea for developing theWENO schemes.
2) A high performance fifth-order multistep WENO (HM-WENO) scheme is developed to improve the accuracy of traditional fifth-order WENO scheme at transition point (connecting a smooth region and a discontinuity point). First, the accuracy of the fifthorder WENO scheme at transition point is analyzed, and then a more effective multistep WENO scheme is developed based on the analysis of the related methods. Theoretical analysis and numerical results show that the new scheme not only improves the accuracy by one order higher than the traditional fifth-order WENO schemes at transition point, but also maintains the fifth-order accuracy in smooth regions even at critical point. The new scheme is simple, which not only improves the computational efficiency of the original multistep WENO scheme, but also avoids the problem of reducing the computational accuracy of other multistep WENO schemes near discontinuities.
3) This paper proposed a low dissipation and low dispersion third-order WENO scheme.
Compared with higher order WENO scheme, the third-order WENO scheme has the advantages of good robustness and easy boundary treatment. However, the existing thirdorder WENO schemes are still too dissipative. In order to further develop the third-order WENO scheme with better performance, first, the functional relationship between the weights of different third-order WENO schemes and the smoothness indicators of substencils is analyzed. And then, a new low dispersion and low dissipation third order WENO scheme is developed by introducing a new function. The new scheme greatly improves the accuracy of the nonlinear weights approximating the optimal weights. And the low dispersion and low dissipation properties of the new scheme are demonstrated by spectral analysis and numerical results.
4) The application of parabolized Navier Stokes (PNS) equation to direct numerical simulation of supersonic boundary layer transition is explored. The PNS equation not only can reflect the characteristics of boundary layer, but also consider the interaction between viscous and inviscid. On the other hand, the PNS equation eliminates the viscous terms related to the approximate mainstream direction, hence it brings great convenience to numerical discretization. Considering that flow transition usually occurs in the boundary layer, it is undoubtedly of great significance to study whether PNS can correctly simulate supersonic boundary layer transition. In this paper, two different computational conditions, i.e., smooth plate with blowing-suction disturbance and plate with roughness elements, are designed. Then, the direct numerical simulation of PNS equation is carried out by using the high order scheme developed in this paper.The numerical results, which compared with the numerical results of the full NS equation, show that the PNS equation can simulate the transition phenomenon of supersonic flat-plate boundary layer.