Two-Dimensional Modeling of Fractional Advection-Dispersion Equation using Meshfree Local Petrov-Galerkin Numerical Method (Case Study: Athabasca River)

Document Type : Research Article


1 PhD Graduate of Hydraulic Structures, Department of Irrigation and Reclamation Engineering, Faculty of Agricultural Engineering and Technology, University College of Agriculture and Natural Resources, University of Tehran, Karaj, Iran.

2 Associate Professor, Department of Irrigation & Reclamation Engineering, University of Tehran, IRAN

3 Associate Professor, Department of Irrigation and Reclamation Engineering, Faculty of Agricultural Engineering and Technology, University College of Agriculture and Natural Resources, University of Tehran, Karaj, Iran



The study of surface water quality has special importance. Unfortunately, sometimes sewage and industrial effluents are discharged into the river. If the mechanism of transport and diffiusion of pollution in rivers with different geometry is known, it can be planned to reduce the effects of pollution on the general health of human society by raising the issue of water mixing and strengthening the self-purification of rivers. The governing equation over the pollution transport phenomenon in rivers is advection-dispersion equation. This equation is classic and does not have the necessary accuracy in predicting the amount of pollutant concentration in the river; Therefore, this equation must be changed in such a way to have the least error in the simulation. The fractional calculus method is used to accurately investigate the transport of pollutants in the stream. To model the pollutant transport in the river, differential equation must be solved. Meshless method is one of the newest numerical methods in recent years that was able to correct some of the disadvantages of mesh-based methods. Studies show that the most studies have focused on solute transport in steady-state flow regimes and regular cross-sections. This study seeks to provide a comprehensive model for simulating the phenomenon of pollutant transport in rivers with steady and non-uniform flow to eliminate the shortcomings of common models. In this study, the parameters of dispersion coefficients, fractional order derivatives and skewness coefficients were optimized in the longitudinal and transverse directions. The values of dispersion coefficient in the longitudinal and transverse directions were 68 and 2.5 m2/s, respectively. The output of the proposed model was compared with the observational data of the Athabasca River and the Mike21 model for three cross sections. The results showed that the amount of R2 in the Meshless Local Petrovo-Galerkin method increased by an average of 11% compared to the Mike model.


Abdeljawad, T., Atangana, A., Gómez-Aguilar, J. F. and Jarad, F. (2019). On a more general fractional integration by parts formulae and applications. Physica A: Statistical Mechanics and its Applications, 536, 122494.
Atluri, S.N. and Zhu, T. (1998). A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 22(2), 117-127.
Belytschko, T., Gu, L. and Lu, Y.Y. (1994). Fracture and crack growth by element free Galerkin methods. Modelling and Simulation in Materials Science and Engineering, 2(3A), 519.
Blank, L. (1996). Numerical treatment of differential equations of fractional order. Numerical Analysis Report, University of Manchester, Department of Mathematics.
Chapra, S.C. (1997). Surface water-quality modeling (Vol. 1): McGraw-Hill, New York.
Deng, Z.Q., Singh, V.P., and Bengtsson, L. (2004). Numerical solution of fractional advection-dispersion equation. Journal of Hydraulic Engineering, 130(5), 422-431.
Deymevar, S. (2018). Numerical solution of shallow water equations using mesh-free Petrov-Galerkin method. M.Sc. thesis, University of Birjand, Birjand. (In Persian)
Fischer, H.B., List, J.E., Koh, C.R., Imberger, J. and Brooks, N.H. (1979). Mixing in inland and coastal waters. Academic press.
Gholami, Z., Yasi, M., Nazi Ghameshlou, A. and Mazaheri, M. (2021). Numerical solution of advection-dispersion equation using mesh-free Petrov-Galerkin method (case study: Murray Burn river). Water and Wastewater Science and Engineering (JWWSE), 6(3), 47-57. (In Persian)
Huang, Q., Huang, G. and Zhan, H. (2008). A finite element solution for the fractional advection–dispersion equation. Advances in Water Resources, 31(12), 1578-1589.
Li, J., Chen, Y. and Pepper, D. (2003). Radial basis function method for 1-D and 2-D groundwater contaminant transport modeling. Computational Mechanics, 32(1), 10-15.
Li, X. and Xu, C. (2010). Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Communications in Computational Physics, 8(5), 1016.
Lian, Y., Wagner, G.J., and Liu, W.K. (2017). A meshfree method for the fractional advection-diffusion equation. In Meshfree methods for partial differential equations VIII, pp. 53-66, Springer, Cham.
Lin, H. and Atluri, S.N. (2000). Meshless local Petrov-Galerkin(MLPG) method for convection diffusion problems. CMES (Computer Modelling in Engineering & Sciences), 1(2), 45-60.
Lin, Z., Wang, D., Qi, D., and Deng, L. (2020). A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations. Computational Mechanics, 
66(2), 323-350.
Liu, G.-R. (2002), Mesh free methods: moving beyond the finite element method, CRC press. New York Washington, D.C.
Liu, G.R. and Gu, Y.T. (2005). An introduction to meshfree methods and their programming. Springer Science & Business Media.
Mahmoodian Shooshtari, M. (2009). Principles of open channel flow. Shahid Chamran University Press, Ahvaz. (In Persian)
Maleki, F. (2016). Pollutant mixing investigation in River using 2D modelling and proposing practical relationships, M.Sc. thesis, Tarbiat Modares University, Tehran. (In Persian)
Mohtashami, A. (2017). Using Mesh-free method for groundwater flow modeling in unconfined aquifer. M.Sc. thesis, University of Birjand, Birjand. (In Persian)
Podlubny, I. (1999). Fractional differential equations. Mathematics in science and engineering, 198, 41-119.
Putz, G. and Smith, D.W. (2000). Two-dimensional modelling of effluent mixing in the Athabasca River downstream of Weldwood of Canada Ltd., Hinton, Alberta.
Riahi-Madvar, H., Ayyoubzadeh, S.A., Khadangi, E., and Ebadzadeh, M.M. (2009). An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS. Expert Systems with Applications, 36(4), 8589-8596.
Tayebi, A., Shekari, Y., and Heydari, M.H. (2017). A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. Journal of computational physics, 340, 655-669.