Hyperbolic Cell-centered Finite Volume Method for Obtaining Potential Magnetic Field Solutions

Abstract

Abstract A hyperbolic cell-centered finite volume solver (HCCFVS) is 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 the second-order Poisson equation into an equivalent first-order pseudo-time-dependent hyperbolic system. Thus, instead of directly solving the 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, the code is applied to numerically solve the solar potential magnetic field problem. The solutions demonstrate the capability of HCCFVS to adequately handle the solar potential field problem, and thus it can provide a promising method of solving the same problem, except for the spherical harmonic expansion and the iterative finite difference method. Finally, by using the potential magnetic fields from HCCFVS and the spherical harmonic expansion as initial inputs, we make a comparative study on the steady-state solar corona in Carrington rotation 2098 to reaffirm the HCCFVS’s performance. Both simulations show that their modeled results are similar and capture the large-scale solar coronal structures. The average relative divergence errors, controlled by solving the Poisson equation in the projection method with HCCFVS for both simulations, are kept at an acceptable level.