Numerical Study of Magnetohydrodynamic Flows in Astrophysics with Spectral Methods

Disk phenomena appear in many astrophysical systems and interact with almost any kind of objects in the Universe. Understanding the dynamics of disks is one of the most important areas of study in modern astrophysics. Although theoretical study of accretion disks have been carried out for half a century, we have not yet achieved a self-consistent understanding. The major difficulties of accretion theory comes from the non-linearity of magnetohydrodynamics as well as the large degree of freedom in turbulent flows, where analytical studies bring limited progress. This dissertation is about developing new numerical algorithms based on analytical understanding in order to describe the dynamics of accretion disks and to study astrophysical turbulence. I will present a new pseudo-spectral algorithm that we developed to study turbulence as well as the stability and general properties of accretion disks. I will show the strengths of this new algorithm in comparison to other previously developed ones. I will also present the results from applications of these algorithms to several problems involving accretion disks