Among the methods designed to preserve the divergence of the magnetic field (see [401] for a review),
the constrained transport method (CT hereafter) designed by [119
] and first extended to HRSC methods
by [343] (see also [229] 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 [24] 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
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
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