CESE Method For Magnetohydrodynamics(MHD)

Magnetohydrodynamics (MHD) simulation is a significant supplement to thetheory and observation of the process of space plasma, which profoundly enlargesthe knowledge of relative physical phenomenal. One of the key issues of MHDsimulation is the control of the divergence of the magnetic field. If the divergenceof the magnetic field cannot be maintain in a small scope, unphysical solutionswill occur, which leads to erroneous computational result. In this thesis, a newly developed numerical scheme, the space-time conser-vation element and solution element (CESE) method, is applied respectively inorder to solve the ideal MHD equations and resistive MHD equations. Com-pared with regular numerical scheme, the main advantages of CESE are iden-tified as follows:(a)Space and time are unified and treated equally when calcu-lating ?ux.(b)The accuracy of computing is higher at the same number of grids.(c)When extending to multiple dimensions, there is no need to apply traditionaloperator splitting or alternating direction technique. (d)There is no need to ap-ply eigen-analysis method, such as Riemann solver etc. In numerical simulation of ideal MHD, by simulating two examples of vor-tex problem and 2-dimensional Riemann problem, the thesis has examined thefollowing three methods which dealt with the divergence of the magnetic field: 8-wave method, EGLM method and Poisson projection method. The results showthat Poisson projection method is the most preferable among the three meth-ods of dealing with the divergence of the magnetic field. Three forms of EGLMmethod are also able to provide satisfying results. On the contrary, the resultof 8-waves method is the least favourable. The divergence of the magnetic fieldobtained by 8-wave method is even bigger than that of no method of dealing withthe divergence of the magnetic field applied. The inherent principles of aboveresults are discussed in this thesis. In numerical simulation of resistive MHD, CESE method and two of itsimproved methods- CNIS method and high order CESE method -are used to simulate two magnetic reconnection problems in single current sheets. A com-parison between the di?erent divergences of the magnetic field as the results ofapplying above-mentioned methods has been made, which indicates that CNISmethod is more preferable as controlling the divergence of magnetic field. We hope that the conclusions drawn by the thesis may be helpful to establishthe numerical mode of solar wind and the design of algorithm which concernsmagnetic reconnection problems of space physics.