High-Order Numerical Methods for Convection-Diffusion Equations with Unbounded Singularities

具有无界奇点的对流扩散方程的高阶数值方法

• 批准号: 1818467
• 负责人: Yang Yang
• 金额: $ 23.7万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818467&HistoricalAwards=false
• 关键词: Order; Numerical; Methods; Convection; Diffusion

This project focuses on design of efficient and robust numerical methods to solve partial differential equations with unbounded singularities, with potential applications in astrophysics, biology, combustion, electrical engineering, and oil recovery. The numerical techniques under development can be extended to systems with multi-species fluid mixtures such as gaseous detonation and reacting flows. The project includes training of graduate students through involvement in the research. Research results will be integrated into new courses on numerical analysis and multidisciplinary computation.The focus of this project is the study of high-order numerical methods for solving convection-diffusion equations with unbounded singularities. It contains two parts. The first part is to study the error behaviors of the numerical schemes and ensure the boundedness of numerical approximations before blow-up occurs (where the exact solutions are sufficiently smooth). The second part is to use high-order numerical methods to solve convection-diffusion equations involving delta-singularities and other blow-up solutions. Special bound-preserving techniques will be constructed to ensure that the numerical approximations are physically relevant. The strategies in this work do not depend on the maximum principle, and they ensure L1-stability of the numerical schemes. For problems with blow-up solutions, the blow-up criteria, blow-up locations, blow-up time, and blow-up rates will be studied. The project aims to develop a general approach to numerically approximate exact blow-up times and to elucidate the relationship between blow-up time and significant system parameters.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.

该项目着重于设计有效且稳健的数值方法，以求解具有无限奇异性的部分微分方程，并在天体物理学，生物学，燃烧，电气工程和石油回收中使用潜在的应用。正在开发的数值技术可以扩展到具有多物种流体混合物（例如气体爆炸和反应流量）的系统。该项目包括通过参与研究来培训研究生。研究结果将纳入有关数值分析和多学科计算的新课程中。该项目的重点是研究以无限奇异性解决对流扩散方程的高级数值方法。它包含两个部分。第一部分是研究数值方案的误差行为，并确保发生爆破之前的数值近似值（在此，确切的解决方案足够光滑）。第二部分是使用高阶数值方法来求解涉及三角线和其他爆炸解决方案的对流扩散方程。将构建特殊的界限技术，以确保数值近似与物理相关。这项工作的策略不取决于最大原则，并且可以确保数值方案的L1稳定性。对于爆炸解决方案的问题，将研究爆炸标准，爆炸位置，爆破时间和爆炸率。该项目旨在开发一种一般方法来在数值上近似确切的爆炸时间，并阐明爆炸时间和重要的系统参数之间的关系。该奖项反映了NSF的法定任务，并通过基金会的智力优点和更广泛的影响审查标准，认为通过评估值得支持。

项目成果

High-Order Bound-Preserving Finite Difference Methods for Incompressible Wormhole Propagation

• DOI: 10.1007/s10915-021-01619-4
• 发表时间: 2021-08
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Xinyuan Liu;Yang Yang-Yang;Hui Guo

High-order local discontinuous Galerkin method for simulating wormhole propagation

• DOI: 10.1016/j.cam.2018.10.021
• 发表时间: 2019-04
• 期刊: J. Comput. Appl. Math.
• 作者: Hui Guo;Lulu Tian;Ziyao Xu;Yang Yang-Yang;Ning Qi

Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media

• DOI: 10.1016/j.jcp.2022.111240
• 发表时间: 2022-04
• 期刊: J. Comput. Phys.
• 作者: Wenjing Feng;Hui Guo;Yue Kang;Yang Yang-Yang

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

• DOI: 10.1090/mcom/3527
• 发表时间: 2020-03
• 期刊: Math. Comput.
• 作者: Yang Yang-Yang;Xiaofeng Cai;Jing-Mei Qiu

High-order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes

• DOI: 10.1016/j.jcp.2019.03.046
• 发表时间: 2019-08
• 期刊: J. Comput. Phys.
• 作者: Ziyao Xu;Yang Yang-Yang;Hui Guo