  Transient diffusion equations arise in many branches of engineering and
applied sciences (e.g., heat transfer and mass transfer), and are parabolic
partial differential equations. It is well-known that, under certain
assumptions on the input data, these equations satisfy important mathematical
properties like maximum principles and the non-negative constraint, which have
implications in mathematical modeling. However, existing numerical formulations
for these types of equations do not, in general, satisfy maximum principles and
the non-negative constraint. In this paper, we present a methodology for
enforcing maximum principles and the non-negative constraint for transient
anisotropic diffusion equation. The method of horizontal lines (also known as
the Rothe method) is applied in which the time is discretized first. This
results in solving steady anisotropic diffusion equation with decay equation at
every discrete time level. The proposed methodology for transient anisotropic
diffusion equation will satisfy maximum principles and the non-negative
constraint on general computational grids, and with no additional restrictions
on the time step. We illustrate the performance and accuracy of the proposed
formulation using representative numerical examples. We also perform numerical
convergence of the proposed methodology. For comparison, we also present the
results from the standard single-field semi-discrete formulation and the
results from a popular software package, which all will violate maximum
principles and the non-negative constraint.
