  This work discusses the application of an affine reconstructed nodal DG
method for unstructured grids of triangles. Solving the diffusion terms in the
DG method is non-trivial due to the solution representations being piecewise
continuous. Hence, the diffusive flux is not defined on the interface of
elements. The proposed numerical approach reconstructs a smooth solution in a
parallelogram that is enclosed by the quadrilateral formed by two adjacent
triangle elements. The interface between these two triangles is the diagonal of
the enclosed parallelogram. Similar to triangles, the mapping of parallelograms
from a physical domain to a reference domain is an affine mapping, which is
necessary for an accurate and efficient implementation of the numerical
algorithm. Thus, all computations can still be performed on the reference
domain, which promotes efficiency in computation and storage. This
reconstruction does not make assumptions on choice of polynomial basis.
Reconstructed DG algorithms have previously been developed for modal
implementations of the convection-diffusion equations. However, to the best of
the authors' knowledge, this is the first practical guideline that has been
proposed for applying the reconstructed algorithm on a nodal discontinuous
Galerkin method with a focus on accuracy and efficiency. The algorithm is
demonstrated on a number of benchmark cases as well as a challenging
substantive problem in HED hydrodynamics with highly disparate diffusion
parameters.
