Numerical Approximation of Classical and Relativistic Magnetohydrodynamics

Abstract

A wide range of phenomena related to solar physics, astrophysics and laboratory plasmas contain many features of magnetohydrodynamics (MHD). This thesis presents the numerical studies of classical and relativistic MHD models. In classical MHD, the two dimensional ideal MHD and the shallow water MHD equations are numerically investigated. The third model is a one-dimensional special relativistic MHD (SRMHD) system. These MHD systems are formed by coupling the Maxwell's equations with the hydrodynamics equations. The simulation of relativistic MHD flows is more difficult due to flow near the speed of light and because of nonlinear relations between conserved and primitive variables. The proposed numerical methods include the space-time conservation element and solution element (CE/SE) method and the kinetic flux vector splitting (KFVS) scheme. In CE/SE solver, the conservation elements are used for calculating the flow variables only, while a central differencing procedure is used to approximate the derivatives of flow variables. The KFVS scheme is based on the direct splitting of macroscopic flux functions of the system. The MUSCL-type reconstruction technique is used to achieve the second order accuracy of the KFVS method. For the designed MHD codes, the projection method is used to satisfy the divergence-free constraint associated with the magnetic field. For validation and comparison, the central scheme is applied to the investigated models. The case studies of ideal MHD include the MHD Kelvin- Helmholtz instability, the MHD rotor problem, the MHD shock interaction with clouds, the blast wave problem and the very famous Orszag-Tang MHD turbulence problem. In the SRMHD case, the one-dimensional relativistic Riemann problems, the interaction of shock fronts, and the relativistic blast wave problems are studied. In all test cases, the suggested computational techniques produced accurate results efficiently.