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

本项目致力于设计高效和稳健的数值方法来求解具有无界奇异性的偏微分方程，在天体物理、生物、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组分流体混合物的系统，如气态爆轰和反应流。该项目包括通过参与研究来培训研究生。研究成果将被整合到数值分析和多学科计算的新课程中。本项目的重点是求解无界奇异对流扩散方程的高阶数值方法的研究。它包含两个部分。第一部分是研究数值格式的误差行为，并在爆破发生前(其中精确解充分光滑)保证数值逼近的有界性。第二部分是用高阶数值方法求解包含Delta奇异性的对流扩散方程和其他爆破解。将构建特殊的保界技术，以确保数值近似是物理上相关的。本文所采用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。对于具有爆破解的问题，将研究爆破准则、爆破位置、爆破时间和爆破速率。该项目旨在开发一种通用方法来以数字近似准确的炸毁时间，并阐明炸毁时间与重要系统参数之间的关系。该奖项反映了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。

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.