Accretion disks are three-dimensional, turbulent, often self-gravitating, magnetohydrodynamic (MHD) flows, which can be modeled with numerical simulations. In this paper, we present a new algorithm that is based on a spectral decomposition method to simulate such flows. Because of the high order of the method, we can solve the induction equation in terms of the magnetic vector potential and, therefore, ensure trivially that the magnetic fields in the numerical solution are divergence free. The spectral method also suffers minimally from numerical dissipation and allows for an easy implementation of models for subgrid physics. Both properties make our method ideal for studying MHD turbulent flows such as those found in accretion disks around compact objects. We verify our algorithm with a series of standard tests and use it to show the development of MHD turbulence in a simulation of an accretion disk. Finally, we study the evolution and saturation of the power spectrum of MHD turbulence driven by the magnetorotational instability.
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