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The field and current induced in a trapped field magnet during a transient current excitation is complicated by the fact that the conductivity of the material changes wildly depending on both the magnitude of the induced current and the local B field. A mutual inductance approach is presented for solving this coupled problem; it involves discretizing the bulk superconductor into cells and precomputing the inductance coupling matrix of all cells with one another and with the excitation coils. The benefits of this approach are that the numerical simulation is rapid and flexible. The latter is important because the conductivity of the individual cells is dependent on the current density, the magnetic field density magnitude, and the temperature. The proposed approach allows the modeling of a very sharp J-E relationship which has proved problematic using conventional finite element approaches. In addition, a phenomenological model relating the critical current density to the field is introduced for these tests in place of the Kim model. The field simulation is tested with a challenging dwell time experiment in which the trapped field above a bulk superconductor is predicted for various excitation current hold times.