随着航空航天领域的发展，计算流体力学（CFD）逐渐成为研究各类复杂气体流动问题的主要工具，但现有CFD数值方法的计算精度和效率仍有待提高。WENO（加权基本无震荡，Weighted Essentially Non-oscillatory）格式因其具有高阶精度和激波捕捉特性，在CFD领域高精度数值方法的研究中受到学者广泛关注。然而目前WENO类算法仍存在一些不足，比如经典WENO格式在等距模板上的线性权值容易出现负值。因此本文基于不等距模板的思想，构造了映射三角WENO（MUS-TWENO）格式、有限体积映射WENO（MUS-WENO）格式和具有自适应线性权值的三角WENO（TWENO-ALW）格式求解双曲守恒律方程，进而将现存的不等距模板WENO（US-WENO）格式和多分辨率WENO（MR-WENO）格式推广到求解分数阶微分方程。具体内容如下：

首先，设计了一种五阶有限差分映射不等距模板的三角WENO（MUS-TWENO）格式，用于求解双曲守恒律方程、高频振荡问题以及一些低密度、低压力、低能量的极端问题。MUS-TWENO格式在三个不等距模板上构造三个不等次的三角多项式，以避免所有等距模板上都存在强激波或接触间断，其复杂的线性权值可设为和为一的任意正数。为了减小三角多项式空间中线性权值和非线性权值之间的差，我们构造了一种新型映射函数和映射非线性权值。在模拟一些高频振荡问题时，即使在光滑区域的临界点附近，以及在极小的 情况下，该格式也能得到较小的数值误差，并获得最优的五阶收敛精度。大量的数值算例验证了MUS-TWENO格式可以在不引入任何保正方法的情况下可有效求解双曲守恒律方程、高频振荡问题和一些极端问题。

其次，在结构网格上设计了求解双曲守恒律方程的五阶有限体积映射不等距模板WENO（MUS-WENO）格式。新型映射WENO型空间重构是一个四次多项式与两个定义在不等距的中心或偏置空间模板上的线性多项式的凸组合。我们提出了一种新型映射函数和映射非线性权值以减小线性权值和非线性权值之间的差。MUS-WENO格式具有截断误差小、阶数精度高的特点。即使在光滑区域的临界点附近，且在极小的 情况下，该格式也能达到最优的五阶收敛精度，同时抑制强间断区域附近的虚假振荡。与经典的有限体积WENO-JS格式和映射WENO（MWENO）格式相比，MUS-WENO格式的线性权值可以取和为一的任意正数，这大大简化了格式的构造过程。同时我们构造了一种新型改进保正方法求解一些低密度、低压力或低能量的问题。大量的数值算例验证了MUS-WENO格式可以很好地求解一些非稳态问题、稳态问题和极端问题。

然后，设计了一种新型具有自适应线性权值的有限差分多阶三角WENO（TWENO-ALW）格式，用于求解双曲守恒律方程。TWENO-ALW格式只使用两个不等距空间模板上定义的信息，而不需要引入其它模板信息可实现三阶、五阶和七阶精度。这使得格式构造简单，很容易推广到任意高阶精度。同时我们提出了一种自适应线性权值过程，即两个线性权值在和为一的条件下自动调整为任意的正数。这使得线性权值更加灵活，能够适应更复杂的情况。TWENO-ALW格式可以使用更少的空间模板，以较低的计算成本实现高阶精度。大量的数值算例表明了TWENO-ALW格式可以很好地模拟双曲守恒律方程和高频振荡问题。

最后，设计了新型六阶有限差分不等距模板WENO（US-WENO6）格式和采用Lax-Wendroff时间离散的高阶有限差分多分辨率WENO（WENO-LW）格式，用于求解含有不连续解的分数阶微分方程。将阶为 的Caputo分数阶导数分解为弱奇异积分和二阶导数项，利用经典Gauss-Jacobi正交求解弱奇异积分，利用新构造的两类WENO型重构方法逼近二阶导数项。在高阶空间重构过程中，线性权值可以取和为一的任意正数。并且与Runge-Kutta时间离散法相比，Lax-Wendroff时间离散法具有更低的计算成本。数值算例表明了US-WENO6格式和WENO-LW格式良好的性能。

本文也进一步验证了基于不等距模板WENO格式的优良属性。该类格式具有灵活的线性权值，同时可以根据实际问题设计合适大小和数量的小模板，从而达到在光滑处大模板可以保持精度，在间断处小模板可以基本无振荡的目的。

With the development of the aerospace field, computational fluid dynamics (CFD) has gradually become the main tool to study various complex gas flow problems, but the computational accuracy and efficiency of the existing CFD numerical methods still need to be improved. WENO (Weighted Essentially Non-oscillatory, Weighted Essentially Non-oscillatory) schemes have attracted wide attention in the research of high-accuracy numerical methods in the field of CFD because of its high order accuracy and shock wave capturing characteristics. However, there are still many shortcomings in WENO algorithms. For example, the linear weight of the classical WENO scheme on the equal-size stencils is prone to negative values. Therefore, based on the idea of unequal-size stencils, this paper constructs a mapped trigonometric WENO (MUS-TWENO) scheme, finite volume mapped WENO (MUS-WENO) scheme, and trigonometric WENO (TWENO-ALW) schemes with adaptive linear weights to solve hyperbolic conservation law equations. Then we extend the existing unequal-size template WENO (US-WENO) scheme and multi-resolution WENO (MR-WENO) scheme to solve fractional differential equations. The details are as follows:

Firstly, a fifth-order finite difference mapped unequal-sized stencils trigonometric WENO (MUS-TWENO) scheme is designed for solving hyperbolic conservation law equations, high oscillation problems, and some extreme problems containing low density, low pressure, and low energy. The MUS-TWENO scheme constructs three unequal-order trigonometric polynomials on three unequal-size stencils to avoid strong shock waves or contact discontinuities in all unequal-size stencils. Its complex linear weights can be set to any positive sum of one. To reduce the difference between linear weights and nonlinear weights in trigonometric polynomial space, we construct a new mapping function and mapped nonlinear weight. It could get smaller numerical errors and obtain optimal fifth-order convergence accuracy with a tiny even near critical points in smooth regions when simulating some highly oscillatory problems. Many numerical examples have verified that the MUS-TWENO scheme can effectively solve hyperbolic conservation law equations, high oscillation problems, and some extreme problems without introducing any positive-preserving methods.

 Secondly, a fifth-order finite volume mapped unequal-sized stencil WENO (MUS-WENO) scheme is designed to solve hyperbolic conservation law equations on structured meshes. The newly mapped WENO-type space reconstruction is a convex combination of a quartic polynomial and two linear polynomials defined on unequal-sized central or biased spatial stencils. We propose a new mapping function and mapped nonlinear weights to reduce the difference between linear weights and nonlinear weights. MUS-WENO scheme has the characteristics of small truncation error and high-order accuracy. Even near the critical point in the smooth regions, this scheme can achieve the optimal fifth-order convergence accuracy under tiny conditions, while suppressing spurious oscillations near strong discontinuities. Compared with the classical finite volume WENO-JS scheme and mapped WENO (MWENO) scheme, the linear weights of MUS-WENO scheme can be any positive number with the sum of one, which greatly simplifies the construction process of the schemes. And we construct a new modified positivity-preserving method is constructed for solving some problems with low density, low pressure, or low energy problems. Many numerical examples have verified that the MUS-WENO scheme can effectively solve some unsteady-state problems, steady-state problems, and extreme problems.

Then, a new finite difference multi-level trigonometric WENO (TWENO-ALW) scheme with adaptive linear weights is designed for solving hyperbolic conservation law equations. TWENO-ALW scheme only uses the information defined on two unequal-size space stencils and do not need to introduce other stencils information to achieve third, fifth, and seventh-order accuracy. This makes the scheme simple to construct and easy to extend to arbitrarily high-order accuracy. At the same time, we propose an adaptive linear weights process where two linear weights are automatically adjusted to any positive number under the condition of a sum of one. This makes the linear weight more flexible and can adapt to more complex situations. TWENO-ALW schemes can use fewer spatial stencils to achieve higher-order accuracy at a lower computational cost. Many numerical examples show that the TWENO-ALW scheme can well simulate hyperbolic conservation law equations and high oscillation problems.

Finally, a new sixth-order finite-difference unequal-size stencils WENO (US-WENO6) scheme and a higher-order finite-difference multi-resolution WENO (WENO-LW) schemes with Lax-Wendroff time discretization is designed to solve the fractional differential equations that may contain discontinuous solutions. The Caputo fractional derivative of order is decomposed into a weak singular integral and a second derivative term. The weak singular integrals are solved by using the classical Gauss-Jacobi quadrature, and the second derivative is approximated by using two newly constructed WENO-type reconstruction methods. In the process of high-order space reconstruction, the linear weights value can be any positive number with a sum of one. Compared with the Runge-Kutta time discretization method, the Lax-Wendroff time discretization method has a lower computational cost. Numerical examples show that the US-WENO6 scheme and WENO-LW schemes have good performance.

This article further validates the excellent properties of the WENO schemes based on unequal-size stencils. This type of schemes has flexible linear weights and can design small stencils of appropriate size and quantity according to actual problems, thus achieving the goal of maintaining accuracy in smooth areas with a large stencil and no oscillation in discontinuous areas with small stencils.

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