| Tenure Colloquium: Structure-preserving numerical schemes for the compressible Resistive-Hall-MHD system Ignacio Tomas Mathematics and Statistics, Texas Tech University |
In this talk, I present a structure-preserving method for the compressible Resistive Hall magnetohydrodynamics model. The differential operator is split into two parts: a hydrodynamic part consisting of the compressible Euler equations and a magnetic part consisting of a system that couples the Lorentz force to the induction equation. The method uses continuous Lagrange elements for the Euler part and a curl-conforming finite element space for the magnetic part. The hydrodynamic part preserves the positivity of the density and internal energy, the conservation of total energy, and the minimum principle for the specific entropy. Owing to the choice of finite elements, the magnetic part preserves the divergence involution constraint. The magnetic part of the system is solved by the Crank-Nicolson method, which requires using Newton's method. Coercivity estimates for the Jacobian of the corresponding Newton iteration are presented. We introduce a high-order artificial resistivity to improve the conditioning of the nonlinear residual and the invertibility of the corresponding Jacobian. Several challenging benchmarks, including a smooth whistler wave, the Orszag–Tang vortex for comparing resistive MHD with resistive Hall-MHD, and a magnetic reconnection problem, are solved to validate the method's robustness and accuracy.
This presentation is archived in the Mediasite catalog at this url ...
mediacast.ttu.edu/Mediasite/Play/da582ce4ae0a4c9988d3f7e1ff937ecf1d
| Promotion Colloquium: Geometry at the Extremes: Rigidity, Symmetry, and Stability Hung Tran Mathematics and Statistics, Texas Tech University |
Our work above contributes to a global program aimed at generalizing Perelman’s Ricci-flow-with-surgery framework, which resolved the Poincaré Conjecture, to higher dimensions. As an application, I will explain how the Ricci flow deforms a closed Riemannian manifold into a round sphere under a natural curvature assumption in any dimension. That establishes our complete resolution of a landmark conjecture proposed by S. Nishikawa. Next, I will highlight our work on the variational stability of critical points via the Morse index, where we introduce an innovative spectral technique to decouple and analyze multiple physical constraints. Finally, I will outline my recent efforts translating pure geometric theories to applied frontiers, highlighting a project on mesh transformations for tensor networks supported by the U.S. Department of Energy and Los Alamos National Laboratory.
This presentation is archived in the Mediasite catalog at this url ...
mediacast.ttu.edu/Mediasite/Play/235ab27e6ed04f808bb054f9304b3e2e1d