Energy comes in two forms; potential energy and kinetic energy. Energyis stored as potential energy and released in the form of kinetic energy. This process of storage and release is the basic strategy of all energy alternatives in use today. This applies to solar, wind, fossil fuels, and the list goes on. Most of these come in diluted and scarce forms allowing only a portion of the energy to be used, which has prompted the quest for the original source, the Sun.

As early as 1905 in the work by Albert Einstein on the connection between mass and energy, it has been seen theoretically that energy can be extracted from the process of fusing lighter elements into heavier elements. Later, this process of fusion was discovered to be the very source powering the Sun. Almost a century later, the work continues to make thermonuclear fusion energy a reality.

Looking closer at the Sun, we see that it consists of a hot burning gas subject to electromagnetic fields, i.e. a plasma. The plasma in the Sun is contained by the massive gravitational force which allows for fusion to be created in a stable and continuous process. Taking inspiration from the Sun we see that a hot plasma and its containment are key to achieving fusion. The gravitational force is not present on Earth, and creating it artificially is, a sof today, an insurmountable task. Fortunately, the plasma can be contained in another way; with magnetic fields.

The challenges of making fusion a viable energy source are numerous and diverse. To deal with these challenges there are several fields of fusion research; engineering, physics, and numerical analysis. These of course overlap, but serve to illustrate the focus of different groups. This thesis work is focused on the latter two, physics and numerical analysis.

The containment of the plasma in a fusion device is degraded by drift wave turbulence. The turbulence in the plasma occurs on the micro-scale, namely on the scale of particles travelling around the magnetic field lines. The physics behind turbulence and the drift waves responsible is a rich field with many future topics.

Since the micro-turbulence can quickly grow and diffuse plasma throughout the device in a matter of micro-seconds, it becomes a difficult challenge to numerically resolve the turbulence over a longer span of time. The typical confinement times required in a fusion device is on the order of several seconds. Thus, the main focus of this thesis is on developing a numerical method that can effectively resolve the plasma physics over longer time-intervals. To this effect, a Time-Spectral method has been developed that utilizes the advantageous properties of spectral methods to all domains, specifically the temporal domain. The numerical method has been implemented on compressible Navier-Stokes, ideal magnetohydrodynamics (MHD), and a toroidal two-fluid plasma turbulence model called the Weiland model.