This thesis deals with modeling and understanding the processes in the atmosphere of the Sun. The solar climate has implications for Earth's climate and effects many of our systems, for instance the Global Positioning System (GPS). In order to protect those systems, it is of vital importance to understand the underlying processes in the Sun that lead to solar flares and mass ejections.
We model the energy transfer in the solar atmosphere by the equations of magneto-hydrodynamics (MHD) with a gravity source term, and suitable initial and boundary conditions. Since there are no analytical results available for these equations, one has to simulate the solutions. This is usually done by finite volume methods. Standard finite volume methods, however, are found to be unstable in multiple dimensions in space.
Therefore, we developed and implemented new robust and accurate methods for simulating the solutions of the MHD equations.
We apply those methods to a two dimensional model with appropriate steady states. The key to obtain a robust scheme is to balance the gravity source with the numerical flux and the Godunov-Powell source. The resulting high order well-balanced schemes are tested on realistic configurations and are found to resolve the complex physical phenomena quite well.