The ideal magnetohydrodynamic (MHD) equations are one of the most important plasma models. The equations model the dynamics of a perfectly conducting quasi-neutral plasma and provide evolution equations for the macroscopic quantities of mass, momentum, and energy density, as well as the magnetic field. MHD have been used successfully in many plasma physics application areas, including in space weather prediction, astrophysics, as well as in laboratory plasma applications such as flows in tokamaks and stellarators. In this thesis, we focus on the development of high-order numerical methods for the ideal MHD equations and its applications.In the first part of the thesis, we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal MHD equations. In the proposed methods, we control divergence errors in the magnetic field by using a novel high-order constrained transport approach to solve the magnetic potential equations. The potential equations are solved using a modified version of the FD-WENO scheme developed for Hamilton-Jacobi equations. Special limiters based on artificial resistivity are also introduced to help control unphysical oscillations in the magnetic field. Several two-dimensional and three-dimensional numerical examples are presented to demonstrate the performance of the proposed method. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. In the second part of the thesis, we focus on the problems involving low density and low pressure in the ideal MHD system. A maximum-principle-preserving flux limiter for scalar hyperbolic conservation laws is extended to a novel positivity-preserving limiter for the ideal MHD equations in this portion. The proposed limiter is applied to the ideal MHD schemes proposed in the first part, resulting in a high-order positivity-preserving scheme. The resulting scheme can achieve high-order accuracy, a discrete divergence-free condition and positivity of the numerical solution simultaneously. Compared to the other positivity-preserving limiter in the literature, our limiter has the advantage that there is no extra CFL restriction from the limiting steps. Numerical examples in one dimension, two dimensions and three dimensions are provided to verify the order of accuracy on smooth test problems and to show the performance when the problems involve low density and/or low pressure.
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