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Summary form only given, as follows. This paper will describe an advanced implicit algorithm for parallel supercomputers to model time-dependent magnetohydrodynamics (MHD) in all three dimensions. The algorithm is a finite volume implementation of a Roe-type approximate Riemann solver with central differenced diffusive terms combined with a lower-upper symmetric Gauss-Seidel iterative method. The ideal MHD equations are hyperbolic, but the addition of non-ideal effects such as resistivity and viscosity changes the equation's mathematical form to mixed hyperbolic-parabolic. The approximate Riemann solver is well suited for tracking discontinuities in the plasma that arise due to the hyperbolic nature of the MHD equations, and a nested finite volume method is used to track the diffusive nature of the parabolic terms. An iterative solver provides the freedom to choose large time steps. The algorithm also works well for steady-state solutions. In addition to describing the algorithm and its parallel implementation, applications will be presented.