tokens: magnetogram data in weak-field regions. Figure 1. Input distribution of the test configuration, displayed in longitude–latitude order. This case has m = 200, n = 400, a = π /2 − 0.1, b = π /2 + 0.1, c = 0, and d = 0.4. The input distribution is a field of random numbers distributed between −0.5 and 0.5. Figure 2. Recovered field of plotted vs. the input field of for the centered grid case shown in Figure 1 . By changing the definition of how finite-difference approximations to spatial derivatives are defined, and where different variables are located within the grid, we can improve this behavior dramatically. In the centered grid case, all variables are colocated at the same grid points. By defining the radial magnetic field and its time derivative to lie at the centers of cells, with the electric field components lying on the edges (or “rails”) surrounding the cells, the finite-difference approximations to the derivatives can be made to obey Faraday’s law to floating-point roundoff error. Figure 3 shows the analogous scatterplot shown in Figure 2 , but using the staggered grid definition described above. The relationship is a straight line. Figure 4 shows the difference between the recovered and original values of . Note that the amplitude of the error is multiplied by 1 × 10 12 , so that the error term is visible in the scatterplot. These two plots clearly show that solutions for the electric fields in a staggered grid formulation can do a far better job of representing the observed data than can the centered grid formulation. Figure 3. Recovered field of plotted vs. the input field of for the staggered grid test case. Figure 4. Difference between recovered and input as a function of the input for the staggered grid case. These figures motivate the development of our more detailed staggered grid formalism, which is described in Section 3.4 . 3.4. The Staggered Grid Formulation for PDFI_SS In three dimensions, Yee ( 1966 ) worked out a second-order-accurate finite-difference formulation for Maxwell’s equations, pointing out that if one places different variables into different locations within the grid, the governing continuum equations (the curls in Maxwell’s equations) become conservative when written down in a finite-difference form. In recent years, “mimetic methods” have been developed, which are higher-order analogs to the Yee grid, in that different variables are defined at different locations within a voxel (such as at interiors, faces, or edges), and with some internal structure in these voxel subdomains allowed. The locations of the variables depend on which integral conservation law is being applied (see, e.g., Candelaresi et al. 2014 and references therein). For our work, the second-order-accurate Yee grid is sufficient. The Yee grid is the basis for the numerical implementation of the MF code described by Cheung DeRosa ( 2012 ). In PDFI_SS, we have a slightly different situation, where most of the calculations are defined on a
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