High Order Multi-Scale Numerical Methods for All-Mach Number Flows

本课题的主要目标是设计高精度、多尺度的全速度流数值模拟方法。据我们所知，高精度的全马赫数流的多尺度求解方法还不发达。拟议的工作预计将与现有的最先进的可压缩流动（激波捕获技术）和不可压缩流动（投影和散度清理技术）的计算工具建立密切的联系。全速度流动模拟方法的成功发展将对计算流体动力学（CFD）相关的广泛应用产生广泛的影响。在算法设计和分析方面的研究工作将促进/受益于CFD应用领域的需求。进一步的影响来自拟议的研究、出版物、参加和组织会议的小型专题讨论会部分以及培训学生的多学科性质。提出了全马赫数流动的高阶多尺度求解方法。PI试图开发一个统一的框架来建立高阶解算器，该解算器可以有效地捕获可压缩欧拉系统的无数值振荡的冲击，并且可以在不需要解析声波的情况下，通过适当的散度清理步骤成功地近似低马赫流动的系统的不可压缩解。特别是，主要的焦点将是有限差分格式与加权基本非振荡（WENO）重建耦合适当的隐式-显式（IMEX） Runge - Kutta （RK）处理，具有以下性质：(1)当马赫数为1阶时，该格式可以鲁棒地捕获可压缩区域的激波前缘；(2)当马赫数趋近于0时，该格式自动成为不可压缩欧拉系统的高阶稳定一致解；(3)在声波高分辨和低分辨情况下，该方案在空间和时间上都具有高阶精度。沿着这个方向，PI在方法论开发、稳定性和渐近保持分析以及广泛的全马赫数流基准测试方面制定了一个全面的计划。进一步扩展到纳维-斯托克斯系统与附加考虑粘性项和特别关注的边界条件将被探讨。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The primary goal of this project is the design of highly accurate and multi-scale numerical methods for all-speed flow simulations. To the best of our knowledge, highly accurate multi-scale solvers for all-Mach number flows are still underdeveloped. The proposed work is expected to establish close connections on existing state-of-art computational tools for compressible flow (with shock capturing techniques) and incompressible flows (with projection and divergence cleaning techniques). Successful methodology developments for all-speed flow simulations will have a broad impact for a wide range of computational fluid dynamics (CFD)-related applications. The proposed research work in algorithm design and analysis will promote/benefit from the demands from CFD applications fields. Further impact comes from the multidisciplinary nature of the proposed research, publications, participation and organization of mini-symposium sections in conferences, as well as the training of students.High order multi-scale solvers for all-Mach number flows are proposed. The PI seeks to develop a uniform framework to build up high order solvers that could effectively capture shocks without numerical oscillations for the compressible Euler system, and would successfully approximate the incompressible solutions of the system with proper divergence cleaning steps for low Mach flows without the need in resolving acoustic waves. In particular, the main focus would be finite difference schemes with weighted essentially non-oscillatory (WENO) reconstructions coupled with proper implicit-explicit (IMEX) Runge Kutta (RK) treatments with the following properties: (1) the schemes can robustly capture shock fronts in the compressible regime when the Mach number is of order 1; (2) the schemes automatically become high order, stable and consistent solvers for the incompressible Euler system when the Mach number approaches 0; (3) the schemes are high order accurate in both space and in time both when the acoustic waves are well-resolved and are under-resolved. Along this direction, the PI develops a thorough plan in methodology development, stability and asymptotic preserving analysis, as well as extensive benchmarked tests for all-Mach number flows. Further extensions to the Navier-Stokes system with additional consideration of viscous terms and special focus on boundary conditions will be explored.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.