Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T07:41:13.298Z Has data issue: false hasContentIssue false
  • English
  • Français

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:knakshatrala@uh.edu; Phone: 713-743-4418 (K. B. Nakshatrala)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Gmsh: A three-dimensional finite element mesh generator with pre- and post-processing facilities. URL: http://www.geuz.org/gmsh/.Google Scholar
[2] Tecplot 360: User's Manual. URL: http://www.tecplot.com, Bellevue, Washington, USA, 2008.Google Scholar
[3]General Algebraic Modeling System (GAMS). Version 23.8, GAMS Development Corporation, Washington DC, USA, 2012.Google Scholar
[4]MATLAB 2012a. The MathWorks, Inc., Natick, Massachusetts, USA, 2012.Google Scholar
[5]Adams, B. M., Bohnhoff, W. J., Dalbey, K. R., Eddy, J. P., Eldred, M. S., Gay, D. M., Haskell, K., Hough, P. D., and Swiler, L. P.. DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 5.2 User's Manual. Sandia Technical Report SAND2010-2183, 2011.Google Scholar
[6]Ascher, U. M. and Petzold, L. R.. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, 1998.Google Scholar
[7]Berman, A. and Plemmons, R.J.. Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.Google Scholar
[8]Berzins, M.. Modified mass matrices and positivity preservation for hyperbolic and parabolic PDEs. Communications in Numerical Methods in Engineering, 17:659–666, 2001.Google Scholar
[9]Bornemann, F. A.. An adaptive multilevel approach to parabolic equations I. General theory and 1D implementation. Impact of Computing in Science and Engineering, 2:279–317, 1990.Google Scholar
[10]Boyd, S. and Vandenberghe, L.. Convex Optimization. Cambridge University Press, Cambridge, UK, 2004.CrossRefGoogle Scholar
[11]Carslaw, H. S. and Jaeger, J. C.. Conduction of Heat in Solids. Oxford University Press, New York, second edition, 1986.Google Scholar
[12]Cattaneo, C.. Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée. Comptes Rendus, 247:431–433, 1958.Google Scholar
[13]Chapko, R. and Kress, R.. Rothe's method for the heat equation and boundary integral equations. Journal of Integral Equations and Applications, 09:47–69, 1997.Google Scholar
[14]Chen, C. M. and Thomee, V.. The lumped mass finite element method for a parabolic problem. Journal of the Australian Mathematical Society, 26:329–354, 1985.Google Scholar
[15]Christie, I. and Hall, C.. The maximum principle for bilinear elements. International Journal for Numerical Methods in Engineering, 20:549–553, 1984.Google Scholar
[16]Chung, J. and Hulbert, G. M.. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. Journal of Applied Mechanics, 60:371–375, 1993.CrossRefGoogle Scholar
[17]Ciarlet, P. G. and Raviart, P-A.. Maximum principle and uniform convergence for the finite element method. Computer Methods in Applied Methods and Engineering, 2:17–31, 1973.Google Scholar
[18]Crank, J.. The Mathematics of Diffusion. Oxford University Press, New York, second edition, 1980.Google Scholar
[19]Douglas, J. and Dupont, T.. Galerkin methods for parabolic equations. SIAM Journal on Numerical Analysis, 07:575–626, 1970.Google Scholar
[20]Elshebli, M. A. T.. Discrete maximum principle for the finite element solution of linear non-stationary diffusionreaction problems. Applied Mathematical Modeling, 32:1530–1541, 1998.Google Scholar
[21]Evans, L. C.. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 1998.Google Scholar
[22]Farago, I., Horvath, R., and Korotov, S.. Discrete maximum principle for linear parabolic problems solved on hybrid meshes. Applied Numerical Mathematics, 53:249–264, 2005.Google Scholar
[23]Fiedler, M.. Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986.Google Scholar
[24]Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order. Springer, New York, 2001.Google Scholar
[25]Gurtin, M. E. and Pipkin, A. C.. A general theory of heat conduction with finite speed. Archive for Rational Mechanics and Analysis, 31:113–126, 1968.Google Scholar
[26]Hairer, E. and Wanner, G.. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, New York, 1996.Google Scholar
[27]Hairer, E., Lubich, C., and Roche, M.. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics. Springer-Verlag, New York, 1989.Google Scholar
[28]Harari, I.. Stability of semidiscrete formulations for parabolic problems at small time steps. Computer Methods in Applied Mechanics and Engineering, 193:1491–1516, 2004.Google Scholar
[29]Herrera, P. and Valocchi, A.. Positive solution of two-dimensional solute transport in heterogeneous aquifers. Ground Water, 44:803–813, 2006.Google Scholar
[30]Horvath, R.. Sufficient conditions of the discrete maximum-minimum principle for parabolic problems on rectangular meshes. International Journal of Computers and Mathematics with Applications, 55:2306–2317, 2008.Google Scholar
[31]Huang, W.. Sign-preserving of principal eigenfunctions in P1 finite element approximation of eigenvalue problems of second-order elliptic operators. Journal of Computational Physics, 274:230–244, 2014.Google Scholar
[32]Huang, W., Kamenski, L., and Lang, J.. Stability of explicit Runge-Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes. WIAS Preprint No. 1869, 2013.Google Scholar
[33]Hughes, T. J. R.. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1987.Google Scholar
[34]Ignaczak, J. and Starzewski, M. O.. Thermoelasticity with Finite Wave Speeds. Oxford Science Publications, New York, 2009.Google Scholar
[35]Ilinca, F. and Hetu, J. F.. Galerkin gradient least-squares formulations for transient conduction heat transfer. Computer Methods in Applied Mechanics and Engineering, 191:3073–3097, 2002.Google Scholar
[36]Jansen, K. E., Whiting, C. H., and Hulbert, G. H.. A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Computer Methods in Applied Mechanics and Engineering, 190:305–319, 2000.Google Scholar
[37]Lang, J. and Walter, A.. An adaptive Rothe method for nonlinear reaction-diffusion systems. Applied Numerical Mathematics, 13:135–146, 1993.Google Scholar
[38]Levi, E. E.. Sull’ equazione del calore. Annali di Matematica Pura ed Applicata, 14:187–264, 1908.Google Scholar
[39]Li, X. and Huang, W.. Maximum principle for the finite element solution of time-dependent anisotropic diffusion problems. Numerical Methods for Partial Differential Equations, 29:1963–1985, 2013.Google Scholar
[40]Lipnikov, K., Shashkov, M., Svyatskiy, D., and Vassilevski, Y.. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. Journal of Computational Physics, 227:492–512, 2007.Google Scholar
[41]Lipnikov, K., Svyatskiy, D., and Vassilevski, Y.. Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. Journal of Computational Physics, 228:703–716, 2009.Google Scholar
[42]Lipnikov, K., Svyatskiy, D., and Vassilevski, Y.. A monotone finite volume method for advectiondif fusion equations on unstructured polygonal meshes. Journal of Computational Physics, 229:4017–4032, 2010.Google Scholar
[43]Lipnikov, K., Manzini, G., and Svyatskiy, D.. Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. Journal of Computational Physics, 230:2620–2642, 2011.Google Scholar
[44]Maxwell, J. C.. On the dynamical theory of gases. Philosophical Transactions of Royal Society of London, A157:26–78, 1866.Google Scholar
[45]Mizukami, A.. Variable explicit finite element methods for unsteady heat conduction equations. Computer Methods in Applied Mechanics and Engineering, 59:101–109, 1986.Google Scholar
[46]Mudunuru, M. K. and Nakshatrala, K. B.. A framework for coupled deformation-diffusion analysis with application to degradation/healing. International Journal for Numerical Methods in Engineering, 89:1144–1170, 2012.Google Scholar
[47] COMSOL Multiphysics. Version 4.3 a. COMSOL Inc, Burlington, MA, 2012.Google Scholar
[48]Munson, T., Sarich, J., Wild, S., Benson, S., and McInnes, L. C.. TAO 2.0 Users Manual. Technical Report ANL/MCS-TM-322, Mathematics and Computer Science Division, Argonne National Laboratory, 2012. http://www.mcs.anl.gov/tao.Google Scholar
[49]Nagarajan, H. and Nakshatrala, K. B.. Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids. International Journal for Numerical Methods in Fluids, 67:820–847, 2011.CrossRefGoogle Scholar
[50]Nakshatrala, K. B. and Valocchi, A. J.. Non-negative mixed finite element formulations for a tensorial diffusion equation. Journal of Computational Physics, 228:6726–6752, 2009.Google Scholar
[51]Nakshatrala, K. B., Prakash, A., and Hjelmstad, K. D.. On dual Schur domain decomposition method for linear first-order transient problems. Journal of Computational Physics, 228:7957–7985, 2009.Google Scholar
[52]Nakshatrala, K. B., Mudunuru, M. K., and Valocchi, A. J.. A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and non-negative constraint. Journal of Computational Physics, 253:278–307, 2013.Google Scholar
[53]Nirenberg, L.. A strong maximum principle for parabolic equations. Communications on Pure and Applied Mathematics, 6:167–177, 1953.Google Scholar
[54]Ozisik, M. N.. Heat Conduction. John Wiley & Sons, Inc., New York, second edition, 1993.Google Scholar
[55]Pang, J.-S.. Methods for quadratic programming: A survey. Computers and Chemical Engineering, 5:583–594, 1983.Google Scholar
[56]Pao, C. V.. Nonlinear Parabolic and Elliptic Equations. Springer-Verlag, New York, 1993.Google Scholar
[57]Payette, G. S., Nakshatrala, K. B., and Reddy, J. N.. On the performance of high-order finite elements with respect to maximum principles and the non-negative constraint for diffusion-type equations. International Journal for Numerical Methods in Engineering, 91:742–771, 2012.Google Scholar
[58]Petzold, L.. Differential/algebraic equations are not ODEs. SIAM Journal on Scientific and Statistical Computing, 3:367–384, 1982.Google Scholar
[59]Picone, M.. Maggiorazione degli integrali delle equazioni totalmente paraboliche alle derivate parziali del secondo ordine. Annali di Matematica Pura ed Applicata, 7:145–192, 1929.Google Scholar
[60]Porru, G. and Serra, S.. Maximum principles for parabolic equations. Journal of the Australian Mathematical Society, 56:41–52, 1994.Google Scholar
[61]Le Potier, C.. Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. Comptes Rendus Mathematique, 341:787–792, 2005.Google Scholar
[62]Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations. Springer-Verlag, New York, 1999.Google Scholar
[63]Rank, E., Katz, C., and Werner, H.. On the importance of the discrete maximum principle in transient analysis using finite element methods. International Journal for Numerical Methods in Engineering, 19:1771–1782, 1983.Google Scholar
[64]Reddy, J. N.. An Introduction to the Finite Element Method. McGraw-Hill, New York, third edition, 2005.Google Scholar
[65]Rothe, E.. Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben. Mathematische Annalen, 102:650–670, 1930.Google Scholar
[66]Saad, Y.. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003.Google Scholar
[67]Thomas, H. R. and Zhou, Z.. An analysis of factors that govern the minimum time step size to be used in the finite element analysis of diffusion problems. Communications in Numerical Methods in Engineering, 14:809–819, 1998.Google Scholar
[68]Ye, Y. and Tse, E.. An extension of Karmarkar's projective algorithm for convex quadratic programming. Mathematical Programming, 44:157–179, 1989.Google Scholar
[69]Zienkiewicz, O. C. and Taylor, R. L.. The Finite Element Method : Vol.1. McGraw-Hill, New York, 1989.Google Scholar