  In solar physics, a severe numerical challenge for modern simulations is
properly representing a transition region between the million-degree hot corona
and a much cooler plasma of about 10000 K (e.g., the upper chromosphere or a
prominence). In previous 1D hydrodynamic simulations, the transition region
adaptive conduction (TRAC) method has been proven to capture aspects better
that are related to mass evaporation and energy exchange. We aim to extend this
method to fully multidimensional magnetohydrodynamic (MHD) settings, as
required for any realistic application in the solar atmosphere. Because modern
MHD simulation tools efficiently exploit parallel supercomputers and can handle
automated grid refinement, we design strategies for any-dimensional block
grid-adaptive MHD simulations. We propose two different strategies and
demonstrate their working with our open-source MPI-AMRVAC code. We benchmark
both strategies on 2D prominence formation based on the
evaporation--condensation scenario, where chromospheric plasma is evaporated
through the transition region and then is collected and ultimately condenses in
the corona. A field-line-based TRACL method and a block-based TRACB method are
introduced and compared in block grid-adaptive 2D MHD simulations. Both methods
yield similar results and are shown to satisfactorily correct the
underestimated chromospheric evaporation, which comes from a poor spatial
resolution in the transition region. Because fully resolving the transition
region in multidimensional MHD settings is virtually impossible, TRACB or TRACL
methods will be needed in any 2D or 3D simulations involving transition region
physics.
