Z-by Pinch Mhd Equations In The Numerical Simulation Method

In the dissertation, we adopt the method of splitting physical processes to study numerical simulation methods in the process of forming plasmas. A whole numerical scheme for the formation of plasmas is designed and the numerical methods for each process are discussed in the details. We give the eigenvalues and eigenvectors for the cylindrical symmetric MHD equations in the conservation form and in the nonconservation form, respectively. The Roe matrix in the case of conservation form is also presented. This work is the basis of constructing high order and high resolution schemes. Applying the Roe matrix, we construct the WENO schemes for the cylindrical symmetric ideal MHD equations and carry out numerical experiments which demonstrate the efficiently and correctness of the schemes. Moreover, we study numerical methods for two kinds of diffusion equations, and discuss how to deal with numerically different boundary conditions and the nonlinear diffusion. We present a method of obtaining symmetrical, diagonal matrixs in solving the magnetic diffusion equation by using the implicit schemes. For Dirichlet boundary conditions, our numerical experiments show that the magnetic field concentrates on the outer boundary; while for Robin boundary conditions, we observe that the magnetic field concentrates on the center of the cylinder. As tests of our whole scheme, we numerically study a simplied model of the MHD. The numerical results show that the diffusion of the magnetic field leads to the moving of the fluid, which can influence the position of the initial interface. This dissertation also studies of difference schemes for fluid dynamic equations in the non-Cartesian coordinates, we give an efficient way of approximating solution of the Riemann problem, and applying the PPM schemes, we present a number of numerical experiments such as the one point and two point blast problems. Finally we discuss the FCT schemes, which can be used to practical problems in the non-Cartesian coordinates. The advantages of the central schemes and the SAMR technique is also discussed and addressed.