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

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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.