Comparison Study Of Several Numerical Methods In Computational Magnetohydrodynamics

Magnetohydrodynamic (MHD) simulation is a signi?cant supplement to re-search the plasma phenomena and evolvement, which profoundly enlarges theknowledge of relative physical phenomenal and achieves some unexpected resultsin theory work. Recently, MHD models has been used successfully to simulatemany important space plasma processes for theoretic predicts in geophysics, as-trophysics, engineering, etc. They can ?ll the gaps left by the spatially andtemporally limited observations and overcome the di?culty of engineering ex-periments. But all the arti?cial factors made it be questioned by scientists.Removing the arti?cial factors and improving the accuracy and resolution is oneof main aims of the development of numerical simulation.In this thesis, two kinds of high accuracy and resolution scheme are dis-cussed: the space-time conservation element and solution element (CESE) methodand its two extensions, high order semi-discrete central-upwind scheme as well.The comparison is taken by several classical MHD numerical problems in detail.CESE scheme, CNIS CESE method and high-order CESE method are ?rstly in-troduced and fully applied to MHD modeling. In order to further understand thehigh order scheme, the third order semi-discrete central-upwind scheme in twodimensions is investigated and a modi?ed semi-discrete central-upwind scheme isdeveloped.The CESE method is a second order method, which has been widely appliedin the ?elds of computational ?uid dynamics. Its numerical dissipation increaseswhile the CFL number decreases. In order to overcome this disadvantage, theCNIS CESE method is introduced. Then a high order CESE scheme is con-structed for advancing the accuracy of this kind of scheme. In the comparison,CNIS CESE is superior to the other two schemes. When the CFL number issmall, the CESE and the high order CESE are dissipative and the discontinuitiesare smooth. But CNIS CESE keeps the accuracy and the shocks are sharp. Ac-cording to 2D simulation, the precision of high order CESE scheme does not be improved remarkably, but its calculation is time consuming. Three of these cankeep the numerical error in a satisfiable level and the error does not accumulateviciously. In conclusion, because of the high accuracy, smaller numerical errorand larger range of CFL number, the CNIS CESE has the best results in thecomparison.Semi-discrete central-upwind scheme adopts the ENO-type or WENO-typehigh order reconstruction. In order to suppress the oscillation near the shocks, alimiter is used in the scheme. But sometimes the limiter is negative at some ex-treme points, so a modifled limiter is introduced to keep the positivity of limiterby us. The numerical results show the modifled scheme still has stability, highaccuracy and smaller oscillation. It can resolve the low pressure and disconti-nuities properly and also keep the numerical error of the divergence of magneticfleld from accumulation along with time.These conclusions drawn by here may be helpful to establish the numericalmode in plasma simulation and the design of algorithm which is more suitablefor the problems.