Toward High-order Solar Corona Simulations: A High-order Hyperbolized Poisson Approach for Magnetic Field Initialization

Proper initialization of the solar corona magnetic field is important for easing the iterative process of realistic and efficient global magnetohydrodynamics (MHD) simulations. This study introduces a high-order flux reconstruction (FR) framework for solving the Poisson equation, a necessary step for computing a potential-field source-surface (PFSS) to initialize the magnetic field for global solar corona simulations with MHD. By hyperbolizing the elliptic Poisson equation into a set of hyperbolic equations, we develop an efficient and robust high-order PFSS solver. Our contributions include developing a Q2 (i.e., quadratic) geometrical representation using prismatic elements for the computational domain, which enables high-order mesh generation. Such a hyperbolized Poisson solver effectively relaxes magnetic fields extrapolated from solar magnetograms, producing scalar potential fields that align well with theoretical expectations. Extensive verification was conducted on the high-order FR solver for polynomial orders up to P3, achieving fourth-order spatial accuracy. The hyperbolized solver demonstrates comparable accuracy to reference solutions (both analytical and numerical) while offering efficient performance, particularly on coarser meshes, making it competitive with state-of-the-art low-order finite volume solvers, which are mostly used for solar MHD simulations. The described developments are a milestone for enabling high-order global solar corona simulations on 3D unstructured grids.