一种带有小波噪声指示器和冲击碰撞方案的空间-时间平滑人工粘性方法，第一部分：一维情况
A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: the 1-D case
AI总结：
本文提出了一种高阶数值算法，用于冲击波与壁面的碰撞和反弹，通过改进C方法，引入新的碰撞指示器和基于小波的噪声检测器，提高了WENO算法的精度并有效处理强间断。
摘要：
在本文中，我们扩展了[40]中开发的C方法，用于在单空间维度中非线性守恒律系统中添加局部化、空间-时间平滑的人工粘性。对于气体动力学，C方法将欧拉方程与一个标量反应扩散方程耦合，其解C作为空间-时间平滑人工粘性指示器。本文的目的在于开发一种高阶数值算法用于冲击波与壁面的碰撞和反弹。具体来说，我们通过引入新的碰撞指示器来推广原始的C方法，该指示器在冲击波与壁面碰撞时自然激活。此外，我们实现了新的基于小波的高频率噪声检测器以及高效的局部噪声去除算法。为了测试该方法，我们使用了一种高度简化的WENO离散化方案。我们展示了我们的方案提高了WENO算法的精度，能够处理极强的间断（高达九个数量级），允许冲击波碰撞和反弹，并去除高频噪声。我们讨论了众所周知的“壁面加热”现象的原因，并证明该病理现象可以在C方法的框架内有效处理。该方法在本文的第二部分[41]中被推广到二维空间。
英文摘要：
In this first part of two papers, we extend the C-method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C-method couples the Euler equations to a scalar reaction-diffusion equation, whose solution $C$ serves as a space-time smooth artificial viscosity indicator. The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C-method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known "wall heating" phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C-method. This method is generalized to two space dimensions in the second part of this work [41].
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