Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Systems with Compact Stencils

用于具有紧凑模板的对流主导系统的龙格-库塔不连续伽辽金方法

• 批准号: 2208391
• 负责人: Zheng Sun
• 金额: $ 15.8万
• 依托单位: University of Alabama Tuscaloosa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208391&HistoricalAwards=false
• 关键词: Runge; Kutta; Discontinuous; Galerkin; Methods

The main objective of this project is to systematically develop a novel class of efficient and high order accurate Runge–Kutta (RK) discontinuous Galerkin (DG) methods for convection-dominated problems and the related applications. The new methods feature improved compactness and local structures. They are expected to be more suitable for parallel computing and implicit time marching in computational fluid dynamics simulation. They have potential applications in diverse areas such as meteorology, oceanography, gas dynamics, aircraft design, hydraulic engineering, oil recovery simulation, and so on. The project will also provide research opportunities for graduate and/or undergraduate students interested in computational mathematics and benefit curriculum development in the PI’s department. In more detail, the PI will investigate a novel approach to reduce the stencil size of the traditional RKDG methods, which typically grows with the number of RK stages. The resulting new methods are referred to as the compact RKDG methods. A comprehensive study of the methods will be carried out in the following directions. Firstly, high order compact RKDG methods will be designed for nonlinear hyperbolic conservation laws. Techniques for oscillation control, implicit time marching, and parallel computing will be investigated. Secondly, a rigorous theoretical framework for convergence, stability, and error analysis of the compact RKDG methods will be established. Thirdly, numerical techniques to preserve the solution bounds and investigate their applications to nonlinear hyperbolic systems in multidimensions will be developed. Finally, in addition to purely convection equations, the methods will be extended to convection-diffusion problems for simulation of viscous flow.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.

该项目的主要目的是系统地开发出一种新型的高效且高阶准确的Runge-Kutta（RK）不连续的Galerkin（DG）方法，用于建筑主导的问题和相关应用。新方法具有改善的紧凑性和局部结构。预计它们将更适合在计算流体动力学模拟中进行平行计算和隐式时间。它们在潜水区域中具有潜在的应用，例如气象，海洋学，气体动力学，飞机设计，液压工程，石油回收模拟等。该项目还将为对PI部门的计算数学和福利课程开发感兴趣的研究生和/或本科生提供研究机会。更详细地，PI将研究一种新颖的方法，以减少传统RKDG方法的模板大小，该方法通常随RK阶段的数量而生长。所得的新方法称为紧凑型RKDG方法。对这些方法的全面研究将在以下方向上进行。首先，高级紧凑型RKDG方法将用于非线性双曲线保护法。将研究用于振荡控制，隐式时间进行和平行计算的技术。其次，将建立紧凑型RKDG方法的收敛，稳定性和误差分析的严格理论框架。第三，将开发用于保留溶液界限并研究其在多维分中非线性双曲系统中的应用的数值技术。最后，除了纯粹的会议方程外，这些方法还将扩展到粘性流量的会议扩散问题。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估被认为是宝贵的支持。

项目成果

On Generalized Gauss–Radau Projections and Optimal Error Estimates of Upwind-Biased DG Methods for the Linear Advection Equation on Special Simplex Meshes

• DOI: 10.1007/s10915-023-02166-w
• 发表时间: 2023-03
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Zheng Sun;Y. Xing

Stability of structure-aware Taylor methods for tents

帐篷结构感知泰勒方法的稳定性

• DOI: 10.1090/mcom/3811
• 发表时间: 2023
• 期刊: Mathematics of Computation
• 影响因子: 2
• 作者: Gopalakrishnan, Jay;Sun, Zheng
• 通讯作者: Sun, Zheng