有限差分 WENO（Weighted Essentially Non-oscillatory）格式是一类高精度 激波捕捉格式，在过去的二十年时间里，这类格式十分流行. 目前流行的有限差 分 WENO 格式都是基于 Jiang 和 Shu 提出的通用框架.

WENO 格式自提出以来，在可压缩流动高精度模拟等领域获得了广泛的应 用. 同时，针对在实际应用中发现的不足之处，人们提出了各种 WENO 格式的改 进方法，以提升格式的分辨率、鲁棒性或计算效率. 总体而言，当前 WENO 格式 的研究多表现在具体方法的改进及应用方面，在 WENO 格式的理论基础方面的 研究成果则相对较少.

基于这一观察，本文着重讨论有限差分 WENO 格式的理论基础.

首先，我们注意到了一个有趣的事实，对于近似 1 阶导数的有限差分 WENO 格式而言，构造格式所需要的线性组合一定存在，即使在整个模版上应用的格式 是降低精度的优化格式. 然而，对于近似 2 阶导数的有限差分 WENO 格式而言， 情况就变得有些不一样了. 即使是在均匀网格的环境下，WENO 式线性组合也有 可能不存在. 基于这一观察，我们系统地研究了有限差分格式在怎样的前提条件 下存在所需要的 WENO 式线性组合，并给出了一个充分必要条件. 这里需要注 意的是，这是一个集合里所有格式（而不是一个具体的格式）存在 WENO 式线 性组合的充分必要条件.

基于前面提到的充分必要条件，我们讨论了最近由 Fu 等提出的 targeted essentially non-oscillatory（TENO）格式. 从 TENO 格式的构造过程中受到启发，我 们得到了有限差分 WENO 格式的理想权重公式. 具体说来，我们先得到了均匀 网格上有限差分 WENO 格式的理想权重公式，然后又将这一结果泛化至非均匀 网格上，得到了 WENO 理想权重的通用表达式.

基于均匀网格上有限差分 WENO 格式理想权重的表达式，我们分别对迎风 型和中心型有限差分 WENO 格式的理想权重证明了单调性，并给出了最大理想 权重和最小理想权重的渐近估计. 利用迎风型有限差分 WENO 格式的最大理想 权重出现的位置，结合格式的谱特性，我们提出了高阶精度有限差分 WENO 格 式所蕴含的洋葱式分层/变宽度子模版结构.

最后，基于前面提到的洋葱式结构，我们使用最近流行的机器学习技术，构 造了一个两层的洋葱格式. 该格式为 1 阶精度迎风格式和 3 阶精度迎风格式的加 权平均，权重函数由强化学习技术训练得出. 我们使用洋葱格式计算了 Sod 问题、 Lax 问题和 123 问题的数值解. 从这几个问题的数值解可以看出基于强化学习的 列表洋葱格式结合了 1 阶精度迎风格式和 3 阶精度迎风格式的优点.

Finite difference WENO (Weighted Essentially Non-oscillatory) schemes are a class of shock capturing schemes which have been quite popular in the past two decades. For now, all the popular finite difference WENO schemes are based on the general framework proposed by Jiang and Shu.

Since the WENO scheme was proposed, it has been widely used in fields such as high order accurate numerical simulations for compressible flows. At the same time, in view of the deficiencies found in practical applications, various improved methods of WENO schemes have been proposed to improve the resolution, robustness or computational efficiency. In general, the current research on the WENO schemes mostly reflects the improvement and application of specific methods, while the research results on the theoretical basis of the WENO schemes are relatively fewer.

Based on such an observation, the thesis is focused on theoretical foundations of finite difference WENO schemes.

Firstly, an interesting fact is noticed, that for the finite difference WENO schemes approximating the 1st derivative, the desired linear combination which is needed in the construction of such schemes always exists, even if the scheme on the whole stencil is optimized by dropping its order of accuracy. However, for finite difference schemes approximating the 2nd derivative, the situation becomes somewhat different. The WENOlike linear combination may not exist even in the context of uniform meshes. Based on such an observation, we systematically studied the prerequisites for the existence of WENO-like linear combination for finite difference schemes and gave a necessary and sufficient condition. What needs to be noted here is that this is a necessary and sufficient condition for all schemes in a set rather than for a specific scheme to have the WENO-like linear combination.

Based on the aforementioned necessary and sufficient condition, we discussed the targeted essentially non-oscillatory (TENO) schemes recently proposed by Fu et al. Inspired by the construction procedure of TENO schemes, we got the formula for ideal weights for finite difference WENO schemes. More specifically, we first obtained the ideal weights for finite difference WENO schemes on uniform meshes. And then the result was generalized into non-uniform meshes to obtain the general expression for WENO ideal weights.

Based on the expressions of the ideal weights for finite difference WENO schemes on uniform meshes, we proved the monotonicity of the ideal weights for upwind and central finite difference WENO schemes and gave the asymptotic estimations for the maximum and minimum ideal weights. Employing the position that the maximum ideal weights occur, together with the spectral properties of the schemes, we proposed the onion-like multiple layers/varying-width-stencils structure implied by the high order accurate finite difference WENO schemes.

Finally, based on the onion-like structure mentioned above, we took advantage of the machine learning techniques which are quite popular recently, and constructed an onion scheme which has two layers. This scheme is a weighted average of the 1st order accurate upwind scheme and the 3rd order accurate upwind scheme, and the weight function is obtained using the reinforcement learning technique. We used the onion scheme to compute numerical solutions for the Sod problem, the Lax problem and the 123 problem. From the numerical solutions we can see that the tabulation onion scheme based on reinforcement learning combines the advantages of both the 1st order and the 3rd order accurate upwind schemes.