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In many problems involving astrophysical and laboratory plasmas one must understand the large scale behavior of a plasma configuration, predict the structure and dynamics of its magnetic field and determine conditions for its stability. This talk will be focused on two such applications: the behavior of magnetic arcades in the solar outer atmosphere (corona) and laboratory experiments designed to simulate coronal and other astrophysical plasmas. The large scale behavior of these plasmas can be described using the theory of magnetohydrodynamics (MHD). However, the inherent three-dimensionality of the dynamics and the presence of widely separated spatial and temporal scales in the problems makes it difficult to solve the MHD equations numerically. I will discuss the models constructed to help understand the dynamics of these plasmas, the questions they help answer and the methods that address the numerical challenges of the equations. In particular, I will describe exponential propagation time integrators which help alleviate problems associated with the stiffness of the MHD system.