The word magnetohydrodynamics derives from magneto-, meaning magnetic field, hydro-, meaning water, and dynamics, meaning movement. It describes the interaction of electrically-conducting fluids and electromagnetic fields. The fundamental concept behind magnetohydrodynamics (MHD) is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes. Applications of MHD play a role in many disciplines such as astrophysics, engineering related to liquid metal, and controlled thermonuclear fusion. Due to its significant role in applications, there have been many studies on the MHD systems from both theoretical and practical viewpoints. Designing and analyzing numerical methods to solve this system is in general a challenging task due to the multi-physical nature of the problem. The aim of this project is to develop novel numerical methods for solving MHD systems efficiently and accurately. The project provides opportunities for graduate and undergraduate student involvement in the research.

The governing equations of the MHD model display a nonlinear coupling between the nonstationary Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. With conventional numerical methods, the approximation may converge to a "wrong'' solution when the exact solution has singularities. In addition, for incompressible MHD models, both the velocity and magnetic fields are divergence free, and numerical computations that do not automatically preserve this property can suffer severe errors. This research aims to design robust structure-preserving numerical methods for MHD equations with high order convergent approximations for both smooth and nonsmooth domains. To achieve this goal, the project employs hybridizable discontinuous Galerkin methods (HDG) in view of their high order accuracy, easy implementation, flexible meshing, and local structure preserving for physical moments, etc. Because many applications require repeated simulations to obtain enough statistical data, this research project will also develop state-of-the-art ensemble HDG algorithms and will combine HDG schemes with other model reduction techniques to further enhance efficiency for large scale computations.

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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Yuliya Gorb
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Old Dominion University Research Foundation
United States
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