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We have demonstrated that the Transition Region Adaptive Conduction (TRAC)method permits fast and accurate numerical solutions of the field-alignedhydrodynamic equations, successfully removing the influence of numericalresolution on the coronal density response to impulsive heating. This isachieved by adjusting the parallel thermal conductivity, radiative loss, andheating rates to broaden the transition region (TR), below a global cutofftemperature, so that the steep gradients are spatially resolved even when usingcoarse numerical grids. Implementing the original 1D formulation of TRAC inmulti-dimensional magnetohydrodynamic (MHD) models would require tracing alarge number of magnetic field lines at every time step in order to prescribe aglobal cutoff temperature to each field line. In this paper, we present ahighly efficient formulation of the TRAC method for use in multi-dimensionalMHD simulations, which does not rely on field line tracing. In the TR, adaptivelocal cutoff temperatures are used instead of global cutoff temperatures tobroaden any unresolved parts of the atmosphere. These local cutoff temperaturesare calculated using only local grid cell quantities, enabling the MHDextension of TRAC to efficiently account for the magnetic field evolution,without tracing field lines. Consistent with analytical predictions, we showthat this approach successfully preserves the properties of the original TRACmethod. In particular, the total radiative losses and heating remain conservedunder the MHD formulation. Results from 2D MHD simulations of impulsive heatingin unsheared and sheared arcades of coronal loops are also presented. Thesesimulations benchmark the MHD TRAC method against a series of 1D models anddemonstrate the versatility and robustness of the method in multi-dimensionalmagnetic fields. We show, for the first time, that pressure differences, ...

A fast multi-dimensional magnetohydrodynamic formulation of the transition region adaptive conduction (TRAC) method