Numerical analysis and well-posedness of the Shear Alfvén wave problem
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In recent years, the study of turbulence phenomena in plasma physics has attracted the interest of many researchers in mathematics and physics, due to its deep theoretical challenges as well as its pivotal role in advancing fusion energy applications. Shear Alfvén waves, described by a coupled system of partial differential equations, capture the fastest oscillatory dynamics within the drift-reduced Braginskii equations. The latter model plasma fluids in high-turbulence regimes subject to strongly anisotropic external forces, such as those in magnetic confinement fusion. These Shear Alfvén waves provide a simplified submodel of the Braginskii framework. Our work addresses the numerical analysis and well-posedness of these equations. We first discuss the dispersion relation, which illustrates the unintuitive relationship between the temporal frequency of oscillations with their spatial scales. We prove existence, uniqueness and stability of weak solutions in the natural energy-norm. Additionally, we propose a Galerkin Finite Element formulation and present a-priori error estimates for temporal and spatial discretization. Finally, numerical example computations are being presented to illustrate convergence and energy conservation properties.