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A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations

Published online by Cambridge University Press:  15 January 2016

K. B. Nakshatrala ,
H. Nagarajan  and
M. Shabouei
K. B. Nakshatrala*
Affiliation:
Department of Civil & Environmental Engineering, University of Houston, Houston, Texas 77204-4003, USA
H. Nagarajan
Affiliation:
Department of Mechanial Engineering, Texas A&M University, College Station, TX 77843, USA
M. Shabouei
Affiliation:
Department of Civil & Environmental Engineering, University of Houston, Houston, Texas 77204-4003, USA
*
*Corresponding author. Email address:[email protected]; Phone: 713-743-4418 (K. B. Nakshatrala)
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Abstract

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.

Keywords

Numerical heat and mass transfermaximum principlesnon-negative solutionsanisotropic diffusionmethod of horizontal linesconvex quadratic programmingparabolic PDEs
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 53 - 93
Copyright
Copyright © Global-Science Press 2016 

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