@Article{CiCP-19-53,
author = {K. B. Nakshatrala, H. Nagarajan and M. Shabouei},
title = {A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations},
journal = {Communications in Computational Physics},
year = {2018},
volume = {19},
number = {1},
pages = {53--93},
abstract = {<p style="text-align: justify;">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 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 proposed methodology is based on the method of horizontal lines in which
the time is discretized first. This results in solving steady anisotropic diffusion equation
with decay equation at every discrete time-level. We also present other plausible
temporal discretizations, and illustrate their shortcomings in meeting maximum principles
and the non-negative constraint. The proposed methodology can handle general
computational grids with no additional restrictions on the time-step. We illustrate the
performance and accuracy of the proposed methodology using representative numerical
examples. We also perform a numerical convergence analysis 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.180615.280815a},
url = {http://global-sci.org/intro/article_detail/cicp/11080.html}
}