Locally-implicit Lax-Wendroff discontinuous Galerkin schemes with constrained transport for the ideal magnetohydrodynamic equations
Date
2025-05
Authors
Pelakh, Ian
Major Professor
Advisor
Rossmanith, James
Parshad, Rana
Wu, Zhijun
Songting, Luo
Yan, Jue
Committee Member
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Abstract
The ideal magnetohydrodynamic (MHD) equations are a system of equations describing the behavior of electrically conducting fluids. The MHD system can be written in symmetric hyperbolic form, but only if an additional condition is satisfied: the magnetic field must be divergence-free, $\vec{\nabla} \cdot \vec{B} =0$. There have been many proposed methods for how to modify the system or methods to account for this additional constraint, such as hyperbolic divergence cleaning and constrained transport. This work develops a novel Lax-Wendroff discontinuous Galerkin finite element method to solve the ideal magnetohydrodynamic equations. New limiters are designed to control unphysical oscillations and guarantee density and pressure positivity. Each time step of the resulting numerical method is equipped with a constrained transport step that enforces the divergence-free property on the discrete magnetic field. The resulting method is tested on a series of standard cases for both the compressible Euler and the ideal magnetohydrodynamic equations, demonstrating the order of accuracy of the scheme on smooth problems, as well as the ability of the scheme to prevent unphysical oscillations and positivity violations on examples with shock waves.
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dissertation