Abstract
An unconditionally-positive finite difference (UPFD) and the standard explicit finite difference schemes are compared to the analytical solution of the advection–diffusion reaction equation which describes the exponential traveling wave in heat and mass transfer processes. It is found that although the unconditional positivity of the UPFD scheme, this scheme is less accurate than the standard explicit finite difference scheme. This is because the UPFD scheme contains additional truncation-error terms in the approximations of the first and second derivatives with respect to x, which are evaluated at different moments in time. While these terms tend to zero as the mesh is refined, the UPFD scheme nevertheless remains less accurate than its standard explicit finite difference counterpart. The presented results are important when modeling a heat and mass transfer processes using the investigated advection–diffusion reaction equation. Furthermore, current and future developers of coupled multi-species transport models may draw on the ideas of solutions methods employed in this study to further develop numerical models for various types of coupled multi-species transport problems.
Similar content being viewed by others
References
Shih, T.M.: Numerical Heat Transfer. Springer, Berlin (1984)
Savović, S., Caldwell, J.: Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method. Thermal Sci. 13, 165–174 (2009)
Savović, S., Simović, A., Djordjevich, A.: Explicit finite difference solution of the power flow equation in W-type optical fibers. Opt. Laser Technol. 44, 1786–1790 (2012)
Murray, J.D.: Mathematical Biology I. Springer, Berlin (2002)
Bear, J.: Hydraulics of Groundwater. Dover, Minneola (2007)
Savović, S., Djordjevich, A., Ristić, G.: Numerical solution of the transport equation describing the radon transport from subsurface soil to buildings. Rad. Prot. Dosim. 150, 213–216 (2012)
Savović, S., Djordjevich, A.: Numerical solution for temporally and spatially dependent solute dispersion of pulse type input concentration in semi-infinite media. Int. J. Heat Mass Transf. 60, 291–295 (2013)
Hetrick, D.K.: Dynamics of Nuclear Reactors. University of Chicago, Chicago (1971)
Urošević, V., Nikezić, D.: Radon transport through concrete and determination of its diffusion coefficient. Radiat. Prot. Dosim. 104, 65–70 (2003)
Djordjevich, A., Savović, S.: Numerical solution of the power flow equation in step index plastic optical fibers. J. Opt. Soc. Am. B 21, 1437–1442 (2004)
Chen-Charpentier, B.M., Kojouharov, H.V.: An unconditionally positivity preserving scheme for advection–diffusion reaction equations. Math. Comput. Model. 57, 2177–2185 (2013)
Quang, D.A., Ehrhardt, M.: Adequate numerical solution of air pollution problems by positive difference schemes on unbounded domains. Math. Comput. Model. 44, 834–856 (2006)
Liu, L., Clemence, D.P., Mickens, R.E.: A nonstandard finite difference scheme for contaminant transport with kinetic Langmuir sorption. Numer. Methods Part. D. E. 27, 767–785 (2011)
Anderson, J.D.: Computational Fluid Dynamics. McGraw-Hill, New York (1995)
Mphephu, N.: Numerical Solution of 1-D Convection-Diffusion-Reaction Equation. Master Thesis, University of Pretoria, South Africa, 25 October 2013.
Djordjevich, A., Savović, S.: Solute transport with longitudinal and transverse diffusion in temporally and spatially dependent flow from a pulse type source. Int. J. Heat Mass Transf. 65, 321–326 (2013)
Acknowledgements
The work described in this paper was supported by the Strategic Research Grant of City University of Hong Kong (Project No. CityU 7004600) and by a grant from Serbian Ministry of Education, Science and Technological Development (Agreement No. 451-03-68/2020-14/200122).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Savović, S., Drljača, B. & Djordjevich, A. A comparative study of two different finite difference methods for solving advection–diffusion reaction equation for modeling exponential traveling wave in heat and mass transfer processes. Ricerche mat 71, 245–252 (2022). https://doi.org/10.1007/s11587-021-00665-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-021-00665-2
Keywords
- Advection–diffusion reaction equation
- Exponential traveling wave
- Finite difference schemes
- Heat and mass flow
- Coupled multi-species transport problems