| In recent years,three-dimensional?3D?numerical modeling of coronal interplan-etary process,which is based on the space physics and high-performance computation,has become an important method to simulate and forecast catastrophic space weather.The solar corona is the key region for the space weather events.Therefore,the 3D nu-merical simulation of the corona is significant in both science and application.The background solar wind and coronal magnetic field extrapolation are the fundamental ingredients of the 3D numerical simulation of the corona.In this thesis,with the help of the six-component grid system and finite volume method,we establish a novel 3D magnetohydrodynamic?MHD?model for the solar corona background by employing the rotational hybrid scheme and develop a new hyperbolic cell-centered finite volume solver by utilizing the hyperbolic method to obtain the coronal potential magnetic field solution.1.3D MHD model with rotated-hybrid schemeThe rotated-hybrid scheme is devised by decomposing a cell-face normal vec-tor into two orthogonal directions and combining the Roe solver,a full-wave or com-plete Riemann solver,and the Rusanov solver,an incomplete Riemann solver,into one rotated-hybrid Riemann solver.One of the two orthogonal directions is the upwind di-rection,defined by the velocity-difference vector,and the Rusanov solver is applied in this direction;another direction is obtained by orthogonality and the Roe solver is ap-plied.The velocity-difference vector is of physical significance.If there is a shock at the cell interface,it will propagate in the direction of the velocity-difference vector?normal to shocks and parallel to shears?.So the Rusanov solver used in this direction can stabi-lize the shock wave and improve robustness of the whole scheme,while the less diffusive solver is applied in the other direction to promote accuracy.In order to keep the mag-netic field divergence-free in MHD simulations,we propose two approaches:Locally Solenoidality-Preserving?LSP?and Globally Solenoidality-Preserving?GSP?.Both of them combine the least-squares reconstruction of magnetic field with the divergence-free constraints.LSP is designed to locally maintain the magnetic solenoidality,while GSP constrains the divergence-free property globally.Using this model,we have simulated the two-dimensional MHD vortex problem and the steady-state coronal structures of Carrington rotation?CR?2068.The simulated results of the MHD vortex problem show that this model not only has good robustness,but also can effectively capture the abruptly changed structures,e.g.,shocks.The nu-merical results of CR 2068 reproduce the coronal large-scale observed structures,in-cluding coronal holes,streamers,pseudo-streamers,and high-and low-speed streams.The model well solves the dilemma of considering divergence-free constraints of the magnetic filed,and can obtain the solutions for long-time evolution problem.Both LSP and GSP can effectively eliminate the magnetic field divergence errors,but the effect of GSP is significantly better than that of LSP.GSP can keep the magnetic field divergence errors to a very low level(about 10-9?10-10order of magnitude).2.A new implementation for obtaining potential magnetic field solutionsMagnetic field extrapolation is a commonly-used method to overcome the diffi-culty of unavailable direct measurement of the 3D coronal magnetic field.In this thesis,we present a new implementation of the extrapolation for coronal potential magnetic field.Unlike the previous methods,we first transform the second-order Poisson equa-tion into an equivalent first-order hyperbolic system,and then use the finite volume method to solve the hyperbolic system.We call this method Hyperbolic Cell-centered Finite Volume Solver?HCCFVS?.To preliminarily demonstrate the effectiveness and accuracy of HCCFVS,two test simulations with exact solutions are first performed to verify its second-order convergence.Then,HCCFVS is applied to the potential field ex-trapolation.The obtained results demonstrate that HCCFVS can adequately handle the solar potential field problem,competitively outperforming the spherical harmonic ex-pansion and the iterative finite difference method.Meanwhile,this method also provides a method for eliminating the magnetic field divergence during the process of solving time-dependent MHD.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 of CR 2098.In two tests with different initial inputs,HCCFVS is used to solve the Poisson equation achieved from the projection method to eliminate the magnetic field divergence.Both simulations show that the modeled results are similar and capture the large-scale solar coronal structures and the averaging errors of magnetic divergence-free are as low as 10-4.In summary,the main points can be listed as follows:·With the help of the six-component grid system and finite volume method,we establish a novel 3D MHD model for the solar corona by employing the rotational hybrid scheme.The numerical results of its 3D solar wind background simulation reproduce the coronal large-scale observed structures,including coronal holes,streamers,pseudo-streamers,and high-and low-speed streams;·According to the global property of the magnetic field divergence-free condi-tion,a global least-squares reconstruction that eliminates the divergence in magnetic field reconstruction is proposed.Numerical tests show that this method can keep the magnetic field divergence errors to be a value less than about 10-9,which is almost close to the machine error;·A new hyperbolic cell-centered finite volume solver is proposed to obtain the coronal potential magnetic field solution.Compared with the spherical harmonic ex-pansion method and the finite difference method,HCCFVS has the merit of extrapola-tion solar coronal magnetic field.Meanwhile,this method is very flexible in using more hyperbolic solvers to build similar methods. |
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