Comparative Analysis of Finite  Difference and Crank-nicolson  Schemes in Simulating  Air Pollution  Dispersion

Jimrise Ochwach; Julius Njiru Nyaga; Mark Okongo

doi:doi:10.11648/j.acm.20261504.11

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Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion

Jimrise Ochwach^*

, Julius Njiru Nyaga, Mark Okongo

Published in Applied and Computational Mathematics (Volume 15, Issue 4)

Received: 5 October 2025 Accepted: 18 October 2025 Published: 11 August 2026

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Abstract

Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.

Published in	Applied and Computational Mathematics (Volume 15, Issue 4)
DOI	10.11648/j.acm.20261504.11
Page(s)	123-132
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Advection-diffusion Equation, Atmospheric Pollutants, Boundary Layer, Finite Difference Method, Numerical Simulation

References

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[2]	Organization, W. H., et al. (2021). WHO global air quality guidelines: particulate matter (PM$_{2.5$ and PM$_{10}$), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide.} World Health Organization.
[3]	Ochwach, J. O., Okongo, M. O., & Musundi, S. W. (2018). Modeling the impact of soil porosity on nitrate leaching to groundwater using the advection-dispersion equation. OSR Journal.
[4]	Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric chemistry and physics: From air pollution to climate change. John Wiley & Sons.
[5]	Wallace, J. M., & Hobbs, P. V. (2006). Atmospheric science: An introductory survey. Elsevier Academic Press, p. 92.
[6]	Arya, S. P., et al. (1999). Air pollution meteorology and dispersion. Oxford University Press, p. 310.
[7]	Mohr, P. J., & Taylor, B. N. (2005). Fundamental physical constants. Physics Today, 56.
[8]	Giovangigli, V. (2015). Multicomponent transport in laminar flames. Proceedings of the Combustion Institute, 35(1), 625-637. https://doi.org/10.1016/j.proci.2014.05.058
[9]	Pendolovska, V., Fernandez, R., Mandl, N., Gugele, B., & Ritter, M. (2013). Annual European Union greenhouse gas inventory 1990-2011 and inventory report 2013. European Environment Agency.
[10]	Ochwach, J. O., Musundi, S. W., & Okongo, M. O. (2018). Stability analysis of the modified advection-dispersion model for nitrate leaching into groundwater. Journal of Progressive Research in Mathematics. Retrieved from https://www.academia.edu/download/81577775/1570-Article-5751-3-10-20181005.pdf
[11]	Jimrise, O., Mark, O., & Ochieng, O. (2022). Mathematical modelling and simulation of nitrate leaching into groundwater. Int J Syst Sci Appl Math, 7, 74-84. https://doi.org/10.11648/j.ijssam.20220705.11
[12]	Ott, W. R. (1999). Mathematical models for predicting indoor air quality from smoking activity. Environmental Health Perspectives, 107 (suppl 2), 375-381.
[13]	Morton, K. W., & Mayers, D. F. (2005). Numerical solution of partial differential equations: an introduction. Cambridge university press.
[14]	Wesseling, P., Zijlema, M., Segal, A., & Kassels, C. (1997). Computation of turbulent flow in general domains. Mathematics and computers in simulation, 44(4), 369-385. https://doi.org/10.1016/S0378-4754(97)00048-5
[15]	Tsega, E. G. (2024). Numerical Solution of Two-Dimensional Nonlinear Unsteady AdvectionDiffusion-Reaction Equations with Variable Coefficients. International Journal of Mathematics and Mathematical Sciences. https://doi.org/10.1155/2024/5541066
[16]	Ren, J. et al. (2024). An advection-diffusion equationbased approach to discern the meteorological factor effects on particle concentrations. Atmospheric Research. https://doi.org/10.1016/j.atmosres.2023.107213
[17]	Kafle, J., Adhikari, K. P., & Poudel, E. P. (2024). Air pollutant dispersion using advection-diffusion equation. Nepal Journal of Environmental Science, 12(1), 1-6. https://doi.org/10.3126/njes.v12i1.47531
[18]	Lee, D. (2023). A comparison of RosenbrockWanner and Crank-Nicolson time integrators for atmospheric modelling. Quarterly Journal of the Royal Meteorological Society. https://doi.org/10.1002/qj.4608

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APA Style

Ochwach, J., Nyaga, J. N., Okongo, M. (2026). Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Applied and Computational Mathematics, 15(4), 123-132. https://doi.org/10.11648/j.acm.20261504.11

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ACS Style

Ochwach, J.; Nyaga, J. N.; Okongo, M. Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Appl. Comput. Math. 2026, 15(4), 123-132. doi: 10.11648/j.acm.20261504.11

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AMA Style

Ochwach J, Nyaga JN, Okongo M. Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Appl Comput Math. 2026;15(4):123-132. doi: 10.11648/j.acm.20261504.11

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@article{10.11648/j.acm.20261504.11,
  author = {Jimrise Ochwach and Julius Njiru Nyaga and Mark Okongo},
  title = {Comparative Analysis of Finite  Difference and Crank-nicolson  Schemes in Simulating  Air Pollution  Dispersion},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {123-132},
  doi = {10.11648/j.acm.20261504.11},
  url = {https://doi.org/10.11648/j.acm.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.11},
  abstract = {Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.},
 year = {2026}
}

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TY  - JOUR
T1  - Comparative Analysis of Finite  Difference and Crank-nicolson  Schemes in Simulating  Air Pollution  Dispersion
AU  - Jimrise Ochwach
AU  - Julius Njiru Nyaga
AU  - Mark Okongo
Y1  - 2026/08/11
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.11
DO  - 10.11648/j.acm.20261504.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 123
EP  - 132
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.11
AB  - Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.
VL  - 15
IS  - 4
ER  -

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Author Information

Jimrise Ochwach

Department of Computing and Information Technology, Mama Ngina University College, Kiambu, Kenya

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http://orcid.org/0000-0002-6354-8706
Julius Njiru Nyaga

Department of Physical Science, Chuka University, Chuka, Kenya

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Mark Okongo

Department of Physical Science, Chuka University, Chuka, Kenya

Contact Email

http://orcid.org/0000-0002-4198-9594

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Ochwach, J., Nyaga, J. N., Okongo, M. (2026). Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Applied and Computational Mathematics, 15(4), 123-132. https://doi.org/10.11648/j.acm.20261504.11

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ACS Style

Ochwach, J.; Nyaga, J. N.; Okongo, M. Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Appl. Comput. Math. 2026, 15(4), 123-132. doi: 10.11648/j.acm.20261504.11

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AMA Style

Ochwach J, Nyaga JN, Okongo M. Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion. Appl Comput Math. 2026;15(4):123-132. doi: 10.11648/j.acm.20261504.11

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@article{10.11648/j.acm.20261504.11,
  author = {Jimrise Ochwach and Julius Njiru Nyaga and Mark Okongo},
  title = {Comparative Analysis of Finite  Difference and Crank-nicolson  Schemes in Simulating  Air Pollution  Dispersion},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {123-132},
  doi = {10.11648/j.acm.20261504.11},
  url = {https://doi.org/10.11648/j.acm.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.11},
  abstract = {Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.},
 year = {2026}
}

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TY  - JOUR
T1  - Comparative Analysis of Finite  Difference and Crank-nicolson  Schemes in Simulating  Air Pollution  Dispersion
AU  - Jimrise Ochwach
AU  - Julius Njiru Nyaga
AU  - Mark Okongo
Y1  - 2026/08/11
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.11
DO  - 10.11648/j.acm.20261504.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 123
EP  - 132
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.11
AB  - Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.
VL  - 15
IS  - 4
ER  -

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