CAREER: High Order Structure-Preserving Numerical Methods for Hyperbolic Conservation Laws

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Partial differential equations of the type known as hyperbolic conservation laws have attracted great attention in mathematical, scientific, and engineering communities due to their wide practical applications in modeling physical systems of interest in fluid mechanics, aerodynamics, meteorology, combustion, and other areas. Development of efficient and accurate numerical algorithms for simulation of solutions to conservation laws continues to be a challenging task. Structure-preserving methods, which provide numerical solutions that preserve a certain continuum property of the underlying models exactly, are recently demonstrated to be more efficient with limited computational resources. This project aims to develop a comprehensive framework to understand structure-preserving methods for hyperbolic conservation laws. The work will have a direct impact in many multi-disciplinary application areas, including fluid and gas dynamics, astrophysics, and atmospheric modeling. This project also has significant broader impact through various educational and outreach activities aimed at students at all levels. These activities include a summer camp program that will expose students including underrepresented minorities to the areas of mathematical modeling, computational science, and computational mathematics. Graduate students will also be mentored and trained through planned working group activities. The notion of conservation (of number, mass, energy, momentum) is a fundamental principle that is used to derive hyperbolic conservation laws. Recent study reveals that structure-preserving numerical methods, which either conserve important physical quantities in addition to mass or preserve other properties of the underlying physical problems, are demonstrated to be more accurate and often have a much improved long time behavior. The objective of this project is to establish a detailed study of novel high-order structure-preserving methods for the linear and nonlinear hyperbolic conservation laws arising in various applications, and to educate students at various levels about the potential and challenges of utilizing numerical simulation to solve important practical problems. The PI aims to study structure-preserving numerical methods in the following directions: (i) Energy conserving methods for wave equations; (ii) Asymptotic preserving methods for kinetic equations; (iii) Well-balanced methods for hyperbolic problems with source terms. The activity is planned to include new algorithm development, theoretical numerical analysis, numerical implementation, and practical applications. This project will also provide excellent training opportunities for graduate and undergraduate students interested in computational sciences, and includes an outreach program for high school students.