Abstract:
High-order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes are highly efficient for multidimensional hyperbolic conservation laws, yet they lack inherent mechanisms to preserve the positivity of physical quantities. This limitation is critical in simulations of compressible turbulence and high-Mach flows, where numerical oscillations near strong shocks or vacuum states can lead to negative density or pressure, causing immediate computation failure. While positivity-preserving techniques exist for finite volume and Discontinuous Galerkin (DG) methods, robust high-order solutions for finite difference schemes remain less mature and often require restrictive Courant-Friedrichs-Lewy conditions.
A novel, conservative, and high-order positivity-preserving sweeping procedure is discussed in this talk. The method addresses the significant challenge of extending scalar sweeping techniques to handle nonlinear concave functions of the conserved variables, most notably the pressure in Euler equations. This approach generalizes the scaling limiter of Zhang and Shu within a global sweeping framework. Unlike standard scalar sweeping, which terminates in a single pass, this nonlinear extension may require iterative adjustments; however, numerical evidence suggests only one or two sweeps are needed, even in demanding regimes.
A key feature of the method is its modularity. It functions as a standalone post-processing step that naturally applies to finite difference, finite volume, and DG methods without any modification to the underlying spatial discretization. The robustness of the technique is demonstrated using fifth-order finite difference WENO, showing that it successfully prevents blow-ups under extreme flow conditions, including shock diffraction and strong blast waves. Ongoing efforts are also briefly outlined, including extensions of this framework to implicit schemes for the Navier–Stokes equations and magnetohydrodynamics.
Speaker’s Bio:
Chloe Griffin is a Ph.D. student in the Division of Applied Mathematics at Brown University, advised by Professor Chi-Wang Shu. Her research focuses on high-order numerical methods for hyperbolic conservation laws, with a specific emphasis on positivity-preserving schemes. This work addresses challenges in both explicit and implicit problems and is designed to be versatile across finite difference, finite volume, and Discontinuous Galerkin frameworks. In parallel with her core thesis research, she is collaborating with the Alfred Wegener Institute in Germany to develop multi-resolution WENO methods for global ocean models.
Mrs. Griffin is a National Science Foundation Graduate Research Fellow and holds a B.S. in Mathematics and a B.A. in Biology from Converse University. Her background extends to scientific machine learning through research at the Karlsruhe Institute of Technology, where she worked on robustness metrics for convolutional neural networks and the fine-tuning of numerical methods using artificial neural networks. She is also an alumnus of the Oak Ridge National Laboratory Science Undergraduate Laboratory Internships program, where she previously conducted research on operator splitting schemes within the Computer Science and Mathematics Division.
