Finite volume method for obtaining potential magnetic field solutions

A hyperbolic cell-centered finite volume solver (HCCFVS) is first proposed to obtain the potential magnetic field solutions prescribed by the solar observed magnetograms. By introducing solution gradients as additional unknowns and adding a pseudo-time derivative, HCCFVS transforms second-order Poisson equation into an equivalent first-order as well as pseudo-time-dependent hyperbolic system. Thus, instead of directly solving the second-order Poisson equation, HCCFVS obtains the solution to the Poisson equation by achieving the steady-state solution to this first-order hyperbolic system. The code is established in Fortran 90 with Message Passing Interface parallelization. To preliminarily demonstrate the effectiveness and accuracy of the code, two test cases with exact solutions are first performed. The numerical results show its second-order convergence. Then, we apply the code to the solar potential magnetic field problem that is often approximated analytically as an expansion of spherical harmonics. A comparison between the potential magnetic field solutions demonstrates the capability of our new HCCFVS to adequately handle the solar potential magnetic field problem, and thus it can be used as an alternative to the spherical harmonics approach. Furthermore, HCCFVS, like the spherical harmonics approach, can be used to provide the initial magnetic field for solar corona or solar wind magnetohydrodynamic (MHD) models. Using the potential magnetic field obtained by HCCFVS as input, the large-scale solar coronal structures during Carrington rotation (CR) 2098 have been studied. Meanwhile, HCCFVS automatically deals with the Poisson projection method to keep the magnetic field divergence-free constraint during the time-relaxation process of achieving the steady state. The numerical results show that the simulated corona captures main solar coronal features and the average relative magnetic field divergence error is maintained to be an acceptable level, which again displays the performance of HCCFVS.