Solving the pollution equations by using the Chebyshev super-spectral viscosity method

Document Type : Original Article

Authors

Department of Water Engineering, College of Agriculture, Razi University, Kermanshah, Iran.

Abstract

Objective: The objective of this research is to solve the pollution equation using the Chebyshev spectral method and the Chebyshev super-spectral viscosity method.
 
Method: Spectral and quasi-spectral methods that have been developed in recent years are able to calculate the solution with higher accuracy than analytical and numerical methods using fewer nodes and less time. These methods have not been investigated for solving pollution equations so far. In this study, the solution of pollution equations using the Chebyshev spectral method and the Chebyshev super spectral viscosity method is investigated. The results of these methods will be compared with the analytical solution under different boundary conditions.
 
Results: In this study, the accuracy of the Chebyshev spectral method and the Chebyshev super spectral viscosity method for solving pollution equations in two modes of continuous and stepwise pollutant injection was investigated in comparison with the analytical solution. The results of these two methods in solving the advection equation showed that with increasing the number of points, the numerical results are closer to the analytical solution, and with the same number of points, the Chebyshev super spectral viscosity method has a higher accuracy in calculating the pollution concentration. However, none of the graphs match the analytical results. The mentioned methods were used to solve the advection-diffusion and advection-diffusion-reaction equations in stepwise injection and were compared with the analytical results. The Chebyshev super spectral method with the same number of points has a higher accuracy than the Chebyshev spectral method in solving the advection-diffusion equation. The results of this method are completely consistent with the analytical results. Also, both methods have the same results for solving the advection-diffusion-reaction equation, which have a high accuracy compared to the analytical solution.
 
Conclusions: the results of this study showed that the Chebyshev spectral methods with high accuracy in solving the pollution equations, the number of points required for them is independent of the length of the river. Thus, at a short length, the number of points required is required in a channel or river with a large length

Keywords

Main Subjects

References

Ajdan, Y., Emadi, A., Chabokpour, J., & Daneshfaraz, R. (2018). Evaluation of some empirical relationships in predicting the longitudinal dispersion coefficient. In: 17th National Hydraulic Conference, Faculty of Engineering, Shahrekord University, Shahrekord, Iran. https://civilica.com/doc/811399 (In Persian)
Barros, F.P.J. de, Colbrook, M.J., & Fokas, A.S. (2019). A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line. International Journal of Heat and Mass Transfer, 139, 482-491. https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.018
Carpenter, S.R., Caraco, N.F., Correll, D.L., Howarth, R.W., Sharpley, A.N., & Smith, V.H. (1998). Nonpoint pollution of surface waters with phosphorus and nitrogen, Ecological Applications, 8(3), 559-568. https://doi.org/10.2307/2641247
Chapra, S.C. (1997). Surface water-quality modeling. McGraw-Hill Publications, New York.  https://www.researchgate.net/publication/48447645_Surface_Water-Quality_Modeling
Chen, H., Wu, Y., & Zhao, L. (2019). Application of Finite Volume Method in Groundwater Contamination Simulation. Journal of Hydrology, 577, 123-135. https://www.researchgate.net/publication/344599189
Fardadi Shilsar, M. J., Mazaheri, M., & Mohammadvali Samani, J. (1400). Analytical solution of the pollution transfer equation with variable coefficients in a river using Laplace transform. Journal of Water and Irrigation Management, 11(4), 683-698. https://doi.org/10.22059/jwim.2021.329149.911
Fathi, N., Rahimpour, M., & Rahnama, M. B. (2014). Numerical solution of diffusion equation using finite element method. Third International Conference on Rainwater Catchment Surface Systems, Birjand, Iran. https://civilica.com/doc/919306
Fattahi Chaqabegi, A. (2012). Mathematical model for simulating the transfer and dispersion of pollution in open channels. Master's thesis. Razi University, Kermanshah, Iran. (In Persian)
Gleick, P. H. (2003). Water Use. Annual Review of Environment and Resources, 28, 275-314. https://www.researchgate.net/publication/230557561_Water_use
Jamali, N. (2019). Analysis and comparative study of numerical methods to solve ordinary differential equations with initial value problem.  International Journal of Advanced Research, 7(5), 117-128. http://dx.doi.org/10.21474/IJAR01/9010
Kato, S., Zhang, C., & Kano, M. (2023). Simple algorithm for judging equivalence of differential-algebraic equation systems. Scientific reports, 13(1), 11534. http://dx.doi.org/10.1038/s41598-023-38254-y
Kilic, M. (2021). Water Pollution: Causes, Negative Effects and Prevention Methods. Istanbul Sabahattin Zaim University Journal of the Institute of Science and Technology, 15(4), 129-132. https://dergipark.org.tr/en/download/article-file/1514334
Kumar, A., Singh, R., & Patel, S. (2021). A Comparative Study of Finite Volume and Finite Element Methods for Pollutant Transport Modeling. Water Resources Management, 35(10), 3507-3523. http://dx.doi.org/10.1142/S0219876208001522
 Kumar, D.,  Kumar, N., & Yadav, R. (2022). Analytical solution for transport of pollutant from time-dependent locations along groundwater, Journal of Hydrology, 610, 0022-1694. https://doi.org/10.1016/j.jhydrol.2022.127826
Lapworth, D.J., Baran, N., Stuart, M.E., & Ward, R.S. (2012). Emerging organic contaminants in groundwater: A review of sources, fate and occurrence. Environmental Pollution, 163, 287-303. https://doi.org/10.1016/j.envpol.2011.12.034
Mahmoud, A., Abdelrahman, E., Inc, Mustafa, Abdo, N., & Mobarak, M. (2022). New exact solutions for the reaction-diffusion equation in mathematical physics. Journal of Ocean Engineering and Science. https://doi.org/10.1016/j.joes.2022.05.006
Mahmoudian Shushtari, M. (2009). Principles of Flow in Open Channels. Shahid Chamran University of Ahwaz Publications, Ahvaz, Iran. https://www.gisoom.com/book/1648721 (In Persian)
Meshdgarmeh, N., Mohammadi Vali, S., & Mazaheri, M. (2013). Analytical solution of the pollution transport equation for arbitrary time pattern of point source pollutants using the Green's function method, Hydraulic Journal, 8(4), 13-25. https://doi.org/10.30482/jhyd.2014.6697 (In Persian)
Noye, J. (1987). Numerical Methods for Solving the Transport Equation. North-Holland Mathematics Studies, North-Holland, 145, 195-229. https://doi.org/10.1016/S0304-0208(08)70035-5
Peyret, R. (2002). Spectral Methods for Incompressible Viscous Flow, Springer New York, New York. http://doi.org/10.1007/978-1-4757-6557-1
Sharma, P., (2018). Transport of Contaminants in Groundwater. ISH Journal of Hydraulic Engineering, 26, 1-10. http://doi.org/10.1080/09715010.2018.1438213
Tadmor, E., (1989). Convergence of spectral methods for nonlinear conservation laws. SIAM Journal on Numerical Analysis, 26(1), 30-44. https://www.jstor.org/stable/2157705
Van Genuchten, M. Th., & Alves, W.J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. Journal of Applied Mathematics and Physics, 2(7). https://www.researchgate.net/publication/234255114_Analytical_Solutions_of_One_Dimensional_Convective_Dispersive_Solute_Transport_Equations
Yadav, R.R., Jaiswal, D.K. & Yadav, H.K., (2010). Analytical solution of one dimensional temporally dependent advection-dispersion equation in homogeneous porous media. International Journal of Engineering, Science and Technology, 2(5), 141-148. http://doi.org/10.4314/ijest.v2i5.60133
Advanced Technologies in Water Efficiency
Volume 5, Issue 3
September 2025
Pages 75-91

Files

History

  • Receive Date: 06 April 2025
  • Revise Date: 29 June 2025
  • Accept Date: 31 July 2025

Share

How to cite

Statistics