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.

本项目的主要目的是系统地发展一类求解对流占优问题的新型高效高精度Runge-Kutta(RK)间断Galerkin(DG)方法及其应用。新方法的特点是改进了紧凑性和局部结构。它们有望更适合于计算流体力学模拟中的并行计算和隐式时间推进。它们在气象学、海洋学、气体动力学、飞机设计、水利工程、采油模拟等领域具有潜在的应用前景。该项目还将为对计算数学感兴趣的研究生和/或本科生提供研究机会，并使PI系的课程开发受益。更详细地说，PI将研究一种新的方法来减少传统RKDG方法的模板尺寸，模板尺寸通常随着RK阶段的数量而增加。由此产生的新方法被称为紧致RKDG方法。将从以下几个方向对这些方法进行全面研究。首先，设计了求解非线性双曲型守恒律的高阶紧致RKDG方法。将研究振荡控制、隐式时间推进和并行计算技术。其次，为紧致RKDG方法的收敛、稳定性和误差分析建立了严格的理论框架。第三，将发展数值技术来保持解的边界，并研究它们在多维非线性双曲组中的应用。最后，除了纯粹的对流方程外，这些方法还将扩展到对流-扩散问题，以模拟粘性流动。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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