CAREER: Tailored Entropy Stable Discretizations of Nonlinear Conservation Laws

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The simulation of fluid flow is foundational to many scientific fields, ranging from environmental and aerospace engineering to solar physics. However, next-generation modeling and analysis is computationally challenging using existing tools. Tailored numerical methods have the potential to address such limitations. For example, projection-based reduced order models decrease computational costs associated with many-query scenarios (such as engineering design or uncertainty quantification) by replacing a high fidelity model with a less expensive low-dimensional surrogate. Similarly, high order accurate schemes are particularly effective at resolving fine-scale features in transient vorticular flows. Unfortunately, when applied to the equations of fluid dynamics, these numerical methods experience non-physical instabilities which can cause simulations to fail unexpectedly. The goal of this project is to enable robust and efficient simulations using discretely "entropy stable" schemes. By building fundamental energetic principles directly into a discretization, entropy stable methods retain accuracy while inheriting verifiable properties which improve "out-of-the-box" robustness. Discretely entropy stable high order discretizations for nonlinear conservation laws have seen rapid development over the last 7 years. This project will extend this methodology to three areas where existing approaches are suboptimal or unavailable: (1) high order methods on non-conforming meshes, (2) high order physical-frame discretizations for domain boundaries with fine-scale features, and (3) reduced order modeling. Additionally, the PI will integrate aspects of the proposed research with an educational program aimed at promoting computational science and improving retention among college students and K-12 teachers. Specifically, the PI will (1) design and supervise senior capstone research projects for engineering undergraduates, and (2) organize a summer research program for K-12 teachers centered around numerical modeling and discovery-based learning. The summer research program will also provide graduate students with mentoring experience, and the PI will follow up by partnering with teachers to incorporate concepts from numerical computing into reusable classroom modules.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.