基于非结构网格的高精度CESE格式

In fluid dynamics, sophisticated structures with very small scales (such as turbulent structures) and discontinuities (shocks or contact discontinuities) with steep jumps in physical quantities usually coexist. This brings tremendous challenges for numerical simulations. Firstly, high-order schemes have a significant advantage over low-order schemes in terms of capturing sophisticated structures with small scales, but they are usually less robust than low-order schemes near discontinuities. Secondly, unstructured meshes are more suitable for complex geometries and are more convenient for mesh adaption than structured meshes. Therefore, constructing robust high-order schemes on unstructured meshes becomes a hotspot in computational fluid dynamics. The space-time conservation element and solution element (CESE) method is an algorithm for compressible computational fluid dynamics proposed in the 1990s. It is a one-step scheme with a super compact stencil, and therefore it can be easily implemented on unstructured meshes. After several decades of developments, the second-order CESE schemes were already very mature and have been applied to many areas. However, the studies on high-order CESE schemes were few and far between. In this project, we will utilize the concept of CESE method to develop a class of new robust high-order compact schemes on unstructured meshes. The aim of the project is to provide an alternative way of constructing high-order schemes.

在流体动力学中，小尺度的精细结构（如湍流结构）和间断（激波或者接触间断）经常同时存在，这给数值模拟带来了极大的挑战。首先，高精度格式相对低精度格式在捕捉小尺度的精细结构方面具有较明显的优势，但是在间断附近的稳定性往往不如低阶精度格式。其次，相对结构网格，非结构网格具有适合复杂几何结构、方便网格自适应等突出优点。因此，在非结构网格上构造稳定的高精度数值方法成为了计算流体力学领域的研究热点之一。时空守恒元解元（CESE）算法是上世纪90年代发展起来的一种可压缩计算流体力学算法。该算法是一种高度紧致的一步算法，很容易应用到非结构网格上。经过几十年的发展，二阶精度CESE格式已经非常成熟并成功应用于多个领域。但是，高精度CESE格式的研究还很欠缺。本项目将利用CESE算法基本原理，在非结构网格上发展一类新的稳定的高精度紧致格式，为高精度格式的构造提供新的途径。

在高速流动中，小尺度的精细结构（如湍流结构）和间断（激波或者接触间断）经常同时存在，高效的数值模拟依赖于高精度激波捕捉。有限体积格式是计算流体力学中常用的格式，但是传统的有限体积格式在提高精度的时候需要不断扩大空间模板，导致非结构的高精度格式构造和编程变得困难。本项目在二阶时空守恒元解元（CESE）方法的基础上，提出了一类任意阶精度的加权紧致中心（WCC）格式：在时间方向采用交错网格，在进行时空积分求解单元平均时避免了求解局部黎曼问题；k空间导数用单元顶点的k-1阶导数的中心差分得到；采用Cauchy–Kovalewski方法将空间导数转换为时间导数；针对格式的特点专门设计了一种WENO限制器。该格式具有非常紧凑的空间模板并且时间方向只需要一步就能达到与空间相同的精度，既可以准确捕捉流场的小尺度结构，也可以保证激波计算的尖锐性和无振荡性。此外，项目基于旋转特征分解方法，提出了一种适用于任意高精度格式的高效的多维限制器，将WCC格式扩展到多维结构网格和非结构网格上。该项目的成果为构造求解双曲守恒律的高精度激波捕捉格式提供了新途径。

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