Many codes have been developed to study highly relativistic, magnetized flows around and inside compact objects. Depending on the adopted formalisms, some of these codes evolve the vector potential A, and others evolve the magnetic field B = curl(A) directly. Given that these codes possess unique strengths, it is sometimes desirable to start a simulation using a code that evolves B and complete it using a code that evolves A. Transferring data from one code to another requires an inverse curl algorithm. This dissertation describes two new inverse curl techniques in the context of Cartesian numerical grids: a cell-by-cell method, which scales approximately linearly with the size of the numerical grid, and a global linear algebra approach, which lacks those ideal scaling properties but is generally more robust, e.g., in the context of a magnetic field possessing some nonzero divergence. We demonstrate that these algorithms successfully generate smooth vector potential configurations in challenging special and general relativistic contexts. In addition, we examine the magnetic helicity, which is a measure of the overall twist of a magnetic field configuration. It is defined as the integral of the dot product of A and B over the volume, and it should be conserved as a system evolves. By examining this quantity, we can put further constraints on the physical accuracy of numerical codes
