一类TVD型组合差分方法及其在磁流体数值计算中的应用

A CLASS OF TVD TYPE COMBINED NUMERICAL SCHEME FOR MHD EQUATIONS AND ITS APPLICATION TO MHD NUMERICAL SIMULATION

根据太阳风数值模拟的特点,考虑到算法的质量（收敛速度、稳定性、精度等）,结合磁流体数值计算的特性,对三维球坐标下磁流体动力学（MHD）方程组中的流体部分采用一种修正Lax-Friedrichs差分法而对磁场部分采用MacCormack格式,发展了一类快捷的具有TVD特性的组合数值新方法.作为格式的检验,在一维情况下,将其与PPM格式进行了比较,对一维快慢磁流体激波问题得到了与PPM格式精度相同的结果,然后将其应用到定态太阳风的数值模拟上,在不同等离子体β情形下,可得到理想的太阳风定态结构,为今后将此数值模式应用到具有复杂磁场位型或三维真实太阳风暴的数值模拟研究奠定了基础.

In the past three decades, solar-terrestrial physicists have introduced many kinds of numerical schemes in computational fluid mechanics to magnetohydrody-namic system in order to simulate various phenomena of solar-terrestrial physics. Along with the recent advance of space weather, great attention has been paid to the development of quick convergence and high resolution numerical schemes of MHD system for the purpose of solar wind storm simulation or establishing numerical prediction methods of space weather. The numerical study of solar wind has undergone a transit from its early simulation problems in supersonic and superAlfvenic domain to the recent works from solar surface, interplanetary space to the interaction between solar wind and the earth's magnetosphere. In this paper, according to the characteristics of numerically modeling solar wind, a new numerical scheme of TVD type for magnetohydrodynamic equations in spherical coordinates is proposed by taking into account of the quality such as convergence, stability, resolution. This new MHD model is established by solving the fluid equations of MHD system with a modified Lax-Friedrichs scheme and the magnetic induction equations with MacCormack II scheme for the purpose of developing a combined scheme of quick convergence as well as of TVD property. To verify the validation of the scheme, the propagation of one-dimensional MHD fast and slow shock problem is discussed with the numerical results conforming to the existing results obtained by the piece-wise parabolic method. Under typical physical parameters on the solar surface, using a dipole as initiation, the steady state numerical results for the solar wind flow is reached by time-relaxation approach. This shows that this numerical model has potential application in modeling solar wind of complex magnetic field and realistic solar-interplanetary storms.