Among the methods designed to preserve the divergence of the magnetic field (see  for a review), the constrained transport method (CT hereafter) designed by  and first extended to HRSC methods by  (see also  for a recent discussion), stands as the preferred choice for most groups. We note, however, that for AMR codes based on unstructured grids and multiple coordinate systems (such as those of [20, 286]) there are other schemes, such as projection methods or hyperbolic divergence cleaning, which appear more suitable than the CT method to enforce the magnetic field divergence-free constraint. The projection method involves solving an elliptic equation for a corrected magnetic field projected onto the subspace of zero divergence solutions by a linear operator. More precisely the magnetic field is decomposed into a curl and a gradient as
Correspondingly, the basic idea of the hyperbolic divergence cleaning approach is to introduce an additional scalar field , which is coupled to the magnetic field by a gradient term in the induction equation. The scalar field is calculated by adding an additional constraint hyperbolic equation given by.
The approach followed by the CT scheme to ensure the solenoidal condition of the magnetic field is based on the use of Stokes theorem after the integration of the induction equation on surfaces of constant and , . Let us write the induction equation as
Following  we can obtain a discretized version of Equation (97) as follows. At a given time, each numerical cell is bounded by six two-surfaces. Let us consider the two-surface , defined by and , and the remaining two coordinates spanning the intervals from to , and from to . The magnetic flux through this two-surface is given by
Integrating Equation (97) on the two-surface , and applying Stokes theorem to the right hand side leads to the following equation for details). Indeed, the total magnetic flux through the closed surface formed by the boundary of the numerical cells and enclosing a volume can be calculated as
Any of the Riemann solvers and flux formulae discussed in Section 4.1.2 can be thus used to calculate the quantities needed to advance in time the magnetic fluxes following Equation (100). At each edge of the numerical cell, is written as an average of the numerical fluxes calculated at the interfaces between the faces whose intersection define the edge. Let us consider, for illustrative purposes, . If the indices denote the center of a numerical cell, an edge is defined by the indices . By definition, . Since and , we can express in terms of the fluxes as follows for additional details on the constrained transport method.
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