ADER Method For MHD Numerical Simulation

The solar-terrestrial physics research includes a number of different physics domains: solar corona, inner heliosphere, magnetosphere, ionosphere electrodynamics and so on. The eruption of the space weather is one of the most important contents of the solar-terrestrial physics research. These large scale and complex physics processes from the Sun to Earth are usually solved numerically. The numerical simulation based on the magnetohydrodynamic (MHD) equations is a very useful tool for the research of these complex physic processes. The MHD numerical simulation, which can help fill the gaps left by the spatially and temporally limited observations, has been used successfully to simulate many important solar-terrestrial space physics problems. The study of numerical methods for MHD equations is a fundamental problem in space physics.In this thesis, a newly developed numerical scheme, the Arbitrary accuracy Derivatives Riemann problem (ADER) scheme, is applied respectively in order to solve the one and two dimensional ideal MHD equations. The ADER scheme is a high order numerical scheme based on the concept of finite volume integration. The main method of this scheme is using a Taylor time expansion at the cell interface position, and then creating the Riemann problems which contain the required variables, in this thesis, we use the HLL scheme to compute these Riemann problems. It is very easy to be extended up to any order of space and time accuracy by using this method. Compared with regular numerical scheme, the main characters of ADER are identified as follows: (a) a high-order spatial representation of the data, instead of the piece-wise constant data of the classical Riemann problem; (b) solution of the generalized Riemann problem at each cell interface, instead of the classical piece-wise constant data Riemann problem of the classical Riemann problem; (c) the numerical flux is computed from a time-integral of the solution of the generalized Riemann problem at the interface.In the one dimensional ideal MHD numerical simulations, by simulating two examples of Brio-Wu and Dai-Woodward shock tube problems, the thesis has proved that the ADER scheme can restrain the oscillatory in the HLL scheme when the CFL number is very large. In the two dimensional ideal MHD numerical simulations, by simulating three examples of vortex problem, two-dimensional Riemann problem and blast wave problem, the thesis has examined the situation of the three example problems in dealing with the divergence of the magnetic field. Through the results we can see that, without dealing with the divergence of the magnetic field, the ADER scheme can do a very good job in keeping the divergence of the magnetic field. We hope that the numerical method proposed here may be helpful to design new numerical scheme for the study of solar wind.