Weighted essentially non-oscillatory (WENO) schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions. However, like many other high-order shock capturing schemes, WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave. This is a long-standing difficulty for high-order shock capturing schemes. In recent years, this non-convergence problem has been studied extensively for WENO schemes. Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations, which are at the small local truncation error level but prevent the residue to settle down to machine zero. Several strategies have been proposed to reduce these slight post shock oscillations, including the design of new smoothness indicators for the fifth-order WENO scheme, the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy, and the design of a new type of WENO schemes. With these strategies, the convergence to steady states is improved significantly. Moreover, the strategies are applicable to other types of weighted schemes. In this paper, we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.

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简述高阶WENO格式的Euler方程对稳态解的收敛性

加权的基本非振荡（WENO）方案是一类高阶震荡捕获方案，已被设计并应用于解决许多流体动力学问题，以研究详细的流动结构及其演化。但是，像许多其他高阶震荡捕获方案一样，WENO方案还存在以下问题：如果有强烈的冲击波，它就不容易收敛到稳态解。对于高阶震动捕获方案，这是一个长期的难题。近年来，对于WENO方案已经对该不收敛问题进行了广泛的研究。数值测试表明，不收敛到稳态的关键原因是轻微的后激波振荡，其在较小的局部截断误差水平上，但是可以防止残留物沉降到机器零。已经提出了几种减少这些轻微震动后振荡的策略，包括为五阶WENO方案设计新的平滑度指标，为更高WENO方案的局部特征投影过程开发高阶加权插值精度的顺序，以及新型WENO方案的设计。使用这些策略，可以显着改善稳态的收敛性。此外，该策略适用于其他类型的加权方案。在本文中，我们简要概述了适用于Euler方程的WENO方案的稳态解的收敛性。较高精度的WENO方案的局部特征投影过程中高阶加权插值的开发，以及新型WENO方案的设计。通过这些策略，可以显着改善稳态的收敛性。此外，该策略适用于其他类型的加权方案。在本文中，我们简要概述了适用于Euler方程的WENO方案的稳态解的收敛性。较高精度的WENO方案的局部特征投影过程中高阶加权插值的开发，以及新型WENO方案的设计。使用这些策略，可以显着改善稳态的收敛性。此外，该策略适用于其他类型的加权方案。在本文中，我们简要概述了适用于Euler方程的WENO方案的稳态解的收敛性。