A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations (Communications in Computational Physics, 2018)
Transient diffusion equations arise in many branches of engineering andapplied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differentialequations. It is well-known that these equations satisfy important mathematicalproperties like maximum principles and the non-negative constraint, which have implicationsin mathematical modeling. However, existing numerical formulations forthese types of equations do not, in general, satisfy maximum principles and the non-negativeconstraint. In this paper, we present a methodology for enforcing maximumprinciples and the non-negative constraint for transient anisotropic diffusion equation.The proposed methodology is based on the method of horizontal lines in whichthe time is discretized first. This results in solving steady anisotropic diffusion equationwith decay equation at every discrete time-level. We also present other plausibletemporal discretizations, and illustrate their shortcomings in meeting maximum principlesand the non-negative constraint. The proposed methodology can handle generalcomputational grids with no additional restrictions on the time-step. We illustrate theperformance and accuracy of the proposed methodology using representative numericalexamples. We also perform a numerical convergence analysis of the proposedmethodology. For comparison, we also present the results from the standard single-fieldsemi-discrete formulation and the results from a popular software package, whichall will violate maximum principles and the non-negative constraint.