In this paper, we employ the direct discontinuous Galerkin (DDG) method for the first time to extrapolate the coronal potential magnetic field (PF) with the source surface (SS) and call the developed numerical model as the DDG-PFSS solver. In this solver, the Laplace’s equation is solved by means of the time-dependent method, i.e., introducing a pseudo-time term into the Laplace’s equation and changing the boundary value problem into the initial-boundary value problem. The steady-state solution of the initial-boundary value problem is the solution of the Laplace’s equation to be solved. This formulation facilitates the implementation of the DDG discretization. In order to validate the DDG-PFSS solver, we test a problem with the exact solution, which demonstrates the effectiveness and third-order accuracy of the solver. Then we apply it to the extrapolation for the coronal potential magnetic field. We use the integral GONG synoptic magnetogram of Carrington rotation (CR) 2060 as the boundary condition and achieve the global potential magnetic field solution by the DDG-PFSS solver. The numerical results such as the coronal holes and streamer belts derived from the DDG-PFSS solver are in good agreement with those obtained from the spherical harmonic expansion method. Also, based on the numerical magnetic field and Wang-Sheeley-Arge model, the obtained solar wind speed is found to basically capture the structures of the high- and low-speed streams observed at 1 AU. These results suggest that the DDG-PFSS solver can be seen as a contribution to the numerical methods for obtaining the global potential magnetic field solutions of the solar corona.

The development of an efficient and accurate method for boundary condition treatments is of fundamental importance to data-driven MHD modeling of the global solar corona and solar active region. Particularly, in a 3D spherical wedge-shaped volume, suitable to the numerical study of solar active region, the transverse terms calls for a delicate treatment at the computational domain’s edges and corners, and properly prescribed conditions for boundaries joining regions of different flow properties, so as to take account of the joint effect of incoming and outgoing waves. To provide a solution to the determination of boundary conditions, in this paper a systematic tactics is formulated for handling edges and corners and the prescribed conditions for inner/outer/edge/corner boundaries are proposed through the combination (CBC-ILW) of the time dependent characteristic boundary conditions (CBCs) and the inverse Lax-Wendroff (ILW) procedure. First, a data-driven 3D MHD simulation has been carried out to study the dynamic evolution of the solar corona from 1Rs to 6.7Rs during the period between 16 May and 6 August 2018. The simulated results of the global coronal evolution provide a good comparison with observed coronal images during the period investigated. Then, the validity of 3D MHD-CBC-ILW is verified for a 3D spherical wedge model, by producing almost the same results as those taken out of the global model on a 3D spherical wedge-shaped volume.