Browsing by Author "Guermond, Jean-Luc"

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Advanced numerical methods for multiphysics Magnetohydrodynamics
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/339; National Science Foundation
The objective of this project is to develop innovative numerical methods capable of solving energy-related problems in the context of renewable and alternative energies. The numerical techniques developed in this project will help design grid-scale liquid metal batteries capable of storing large quantities of renewable energies. This research will also help improve the performance of large power electric transformers cooled by environment-friendly vegetable-based oils containing ferromagnetic particles. Finally, by facilitating the understanding of magneto-hydrodynamic instabilities in liquid metals, this project will help to ascertain the integrity and the efficiency of the electromagnetic pumps that will be used to extract energy from the next generation of Liquid-Metal Fast-Breeder Reactors and Tokamaks. This project will be done in collaboration with an European team; the project will foster diversity, international exchanges, and multidisciplinarity. The educational component of the project will contribute to increase the competitiveness of the STEM workforce in the US in computational magnetohydrodynamics. The research program will be organized into four areas: (1) Development of new efficient semi-implicit algorithms to solve partial differential equations with variable material properties (density, electric conductivity, magnetic permeability) using spectral or very high-order methods; (2) Modeling of ferromagnetic fluids and development of new numerical techniques to solve the magneto-static equations in the context of liquid metals and ferromagnetic fluids; (3) Development of level set techniques to account for more than two phases, and development of new high-order level set techniques to guarantee mass conservation and maximum principle; (4) Integration of the mathematical models and numerical techniques developed in (1)-(2)-(3) into a massively parallel open source code to test the proposed methods on realistic applications (liquid metal batteries, thermo-convection of ferromagnetic oil in high-voltage transformers, liquid metal dynamos). This project will involve the Principal Investigator, one post-doctoral collaborator, one graduate student, and European collaborators.
Robust and Accurate Approximation of Hyperbolic Systems
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/339; DOD-Air Force-Office of Scientific Research
The project consists of developing robust numerical methods for solving hyperbolic systems of conservation laws such as the Euler equations, the shallow water equations, and related systems such as radiative hydrodynamics. Robustness is an area of investigation that is not well addressed in the literature. We think that most current high-order methods are not robust. This lack of robustness is what makes them unattractive to practitioners. Real progresses will come when robust accurate methods are readily available to a large community of practitioners. We do not think that this is the case at the moment. Our research program is articulated around the following five points: (i) The numerical method must be invariant domain preserving (i.e. preserve convex invariant domains of the problem) on any unstructured meshes in any space dimension; (ii) The method must be robust in the sense that it should not involve any tuning parameter, mesh-dependent coefficient, or problemdependent stabilization. A robust method should also be easy to program and parallelize. It should not require any subtle mathematical knowledge from practitioners to run it; (iii) The method must be at least third-order accurate in space and time and must be open to higher-order extensions; (iv) The method should respect the physical dissipation of the PDE; (v) The above objectives must be reached by stating precise statements supported either by mathematical proofs or very strong numerical evidences