numerical-mhd-ct.tex

\subsection{Magnetohydrodynamics: Constrained transport}
\label{sec.num.mhd-ct}
\def\Bvec{{\bf B}}
\def\Bf{Bf}
\def\Bc{Bc}
\def\Evec{{\bf E}}
\def\Divb{\ensuremath{\div \Bvec}}

The third solver we describe is an MHD method developed by
\citet{Collins10}.  Since a full description and suite of test
problems can be found in that reference, we only describe the method
briefly here.

The divergence of the magnetic field, \Divb, is identically zero in
reality due to the fact that the evolution of the magnetic field is
the curl of a vector, and the divergence of the curl of a vector is
identically zero.  The Constrained Transport (CT) method
\citep{Evans88, Balsara99} for magnetohydrodynamic (MHD) evolution
employs this same vector property to evolve the magnetic field in a
manner that preserves $\Divb=0$.  The electric field is computed using
the fluxes from the Riemann solver.  The curl of that electric field
is then used to update the magnetic field.  The advantage of this
method is that it preserves \Divb\ to machine precision.  The primary
drawback is increased algorithm complexity.  Note also that since only
the update of the magnetic field is divergence free, any monopoles
created by other numerical sources (such as ill-chosen initial
conditions) persist.

The base Godunov method, described in \citet{Li08a}, uses spatially
and temporally second-order reconstruction (both MUSCL-Hancock and
Piecewise Linear Method), and a selection of Riemann solvers including
HLLC and HLLD \citep{Mignone07}, as described earlier.  The
constrained transport methods are the first-order method described by
\citet{Balsara99} and the second-order methods described in
\citet{Gardiner05}.  The AMR machinery is described by
\citet{Balsara01} and \citet{Collins10}.

The increased complexity of the constrained transport scheme comes in
the form of area-averaged face and length-averaged edge-centered
variables, while the rest of \enzo\ employs predominantly
volume-averaged cell-centered variables.  The magnetic field is
represented by both a face-centered field, B$_f$, and a cell-centered
field, B$_c$.  The electric field is edge-centered.  The magnetic
field is updated in four steps: first, the Riemann problem is solved
in the traditional manner, using the cell-centered field; second, an
edge-centered electric field is computed using the fluxes from the
Riemann solver; third, the curl of that electric field is used to
update the face-centered field; finally, the cell-centered magnetic
field is updated with an average of the face-centered field.

Divergence-free AMR is somewhat more complex than the AMR employed
elsewhere in \enzo.  First, the interpolation must be constrained to
be divergence free.  Thus, all three face-centered field components
are interpolated in concert.  Second, any magnetic field information
from the previous timestep must be included in the interpolation,
making the interpolation more complex than the simple parent-child
relation used for other fields.  Third, the flux correction involves
more possible grid relations than traditional AMR.  In order to
circumvent this last complexity, the electric field is projected from
fine grids to parent grids (rather than the magnetic field), and is
then used to re-update the parent magnetic field.  This is described
in detail in \citet{Balsara99} and \citet{Collins10}.

The dual energy formalism has also been incorporated in two possible
ways -- one that uses internal energy, as described in Section
\ref{sec.hydro.ppm}, and one that uses entropy \cite{TVD93,
Collins10}.