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An Adaptive Mesh Refinement High-Order Flux Reconstruction Method for Shock Capturing and Magnetohydrodynamics Open Access

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Magnetohydrodynamic simulations with complex structures of shocks are often characterized by a wide spectrum of length scales and abundance of flow features. In astrophysical simulations, the complexity and computational cost grow exponentially with the admission of various types of shocks. For example, in space weather forecast, the eruptive solar events and the accompanied network of shocks are prohibitive for many existing solvers. This dissertation research tackles these challenges by developing a high-order adaptive mesh refinement computational framework with robust shock capturing capability.The flux reconstruction (FR) method, featuring high-order accuracy and low numerical dissipation on unstructured grid, is extended to magnetohydrodynamic equations for the first time. To accurately capture all types of shocks/discontinuities and their evolution, interaction and complex structure, an adaptive mesh refinement (AMR) method and an artificial diffusivity/resistivity (AD/AR) are developed for this task. A hyperbolic cleaning method is employed to maintain the divergence-free constraint of the magnetic field, meanwhile the overall conservative, hyperbolic form of governing equations are intact. When facing stringent stability requirement, a decoupled, sub-iterative cleaning method is devised to thoroughly diminish the divergence error to a desired level.The novel data structure of the AMR method is built with direct addressing when retrieving information across the grid hierarchy, on unstructured, even dynamic grids. The high-order accuracy is retained when AMR is activated with the presence of grid motion. The FR-AMR method is also very suitable for prallelization which paves the way for the overall solver to be further developed for large scale computations in the future. The acceleration acquired by using the AMR method shows a linear growth rate, which verifies that the solver is not burdened by the overhead associated with AMR. The coordination of AMR and AD/AR demonstrates accurate tracing of evolving shocks and minimal added numerical diffusion. The robustness and effectiveness of the shock capturing duo are tested on the hydrodynamic system, and extended to the MHD system which admits a number of unique shocks and discontinuities. The sub-iterative hyperbolic cleaning method tackles divergence error in response to various levels of plasma with a relaxed pseudo time stepping and maximum wave speed. The overall computational framework is demonstrated to have high-order accuracy, effective shock capturing capability, efficient solution speed. The multi-scale resolution flexibility and controlled divergence error of the FR-AMR-MHD framework make it a strong candidate in simulating eruptive solar events, such as coronal mass ejection, where the commonly used anelastic assumption and static spherical grid can be remedied.

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