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.

本项目致力于设计有效且稳健的数值方法来求解具有无界奇点的偏微分方程，在天体物理学、生物学、燃烧、电气工程和石油开采等领域具有潜在的应用前景。正在开发的数值技术可以扩展到具有多种流体混合物的系统，如气体爆轰和反应流。该项目包括通过参与研究来培训研究生。研究成果将整合到数值分析和多学科计算的新课程中。本课题的重点是研究求解具有无界奇点的对流扩散方程的高阶数值方法。它包含两个部分。第一部分是研究数值格式的误差行为，并保证数值逼近在爆破发生前（精确解足够光滑时）的有界性。第二部分是用高阶数值方法求解包含δ奇点的对流扩散方程和其他爆破解。将构造特殊的保界技术，以确保数值近似是物理相关的。本工作中的策略不依赖于极大值原理，它们保证了数值格式的l1稳定性。对于有爆破解的问题，将研究爆破标准、爆破地点、爆破时间和爆破率。该项目旨在开发一种通用的方法来数值近似精确的爆破时间，并阐明爆破时间与重要系统参数之间的关系。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。

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

Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation

• DOI: 10.1051/m2an/2021020
• 发表时间: 2021-04
• 期刊: ESAIM: Mathematical Modelling and Numerical Analysis
• 作者: Hui Guo;Rui-yu Jia;Lulu Tian;Yang Yang-Yang

High-order bound-preserving finite difference methods for multispecies and multireaction detonations

多物种和多反应爆炸的高阶保界有限差分法

• DOI: 10.1007/s42967-020-00117-y
• 发表时间: 2021
• 期刊: Communications on Applied Mathematics and Computation
• 影响因子: 1.6
• 作者: Jie Du;Yang Yang
• 通讯作者: Yang Yang