High order entropy stable finite elements for unsteady compressible flows via entropy correction artificial viscosity

The National Science Foundation Division of Mathematical Sciences awarded the University of Texas at Austin $270,000 on January 15, 2027, under the Mathematical and Physical Sciences program to develop entropy-stable high-order finite element numerical methods for unsteady compressible flows.

The project addresses computational bottlenecks in existing high-order structure-preserving methods for computational fluid dynamics by extending entropy-stable schemes to complex and non-ideal thermodynamic regimes—including non-ideal equations of state and thermally perfect gases—and constructing robust and efficient continuous spectral element methods. The work aims to produce simulation tools that are both highly accurate and inherently stable for modeling fluid and gas behavior at high speeds, with applications to aircraft design, advanced manufacturing, and national defense technologies. The project will also train undergraduate students and contribute to the development of high-performance open-source software.

Performance occurs in Austin, Texas through December 31, 2029. No sub-awards are planned.

Abstract

Efficiently simulating how fluids and gases behave at high speeds is critical for designing safe aircraft, improving advanced manufacturing processes, and supporting national defense technologies. However, existing simulation tools often struggle with stability when modeling real-world scenarios or can require oversimplified assumptions that limit their applicability. This project aims to design next-generation computational models that are both highly accurate and inherently stable by ensuring that simulations obey fundamental physical laws. Additionally, the project will train undergraduate students and contribute to the development of high-performance open-source software. This project focuses on the development of entropy-stable numerical methods that enforce a high-order accurate cell entropy inequality using minimally invasive and low-dissipation correction terms. These methods aim to address two challenges in high-order structure-preserving methods for computational fluid dynamics: extending high-order entropy-stable schemes to complex and non-ideal thermodynamic regimes (including non-ideal equations of state and thermally perfect gases), and constructing robust and efficient continuous spectral element methods to address computational bottlenecks associated with discontinuous finite element methods. 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.- subawards are not planned for this award.