Mathematical Based Implicit and Explicit Finite Difference Techniques for Solving the Ground Water Flow Equations Using Spreadsheets

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Farzin Salmasi 1 Mohammad Taghi Sattari 1,* Halit Apaydin 1

1. Department of Water Engineering, Faculty of Agriculture, University of Tabriz, Tabriz 51666 IRAN

* Corresponding author.


Received: 19 Feb. 2022 / Revised: 17 Mar. 2022 / Accepted: 13 May 2022 / Published: 8 Oct. 2022

Index Terms

Critical thinking, explicit, groundwater modelling, finite difference, implicit, spreadsheet


In countries with arid and semi-arid climate such as Iran with water constraints, the use of groundwater resources is very important. There are various mathematical based methods and software packages for modelling groundwater resources. This paper uses groundwater flow problems to illustrate possible approaches for providing the environment of active teaching. Mathematical models supported by software applications facilitate the gaining of an insight into the physical behaviors by investigating a host of scenarios and events but they are poor in training critical thinking for encapsulating the hardcore mathematical equations describing the problems. Whilst software engineering has transformed the intellectual capitals accumulated between the 20th century and the middle of the 21th century into working tools, it has the drawback of encapsulating core mathematics away from common experience of the students and practitioners. This diminishes critical thinking in a world of increasing risks and ought to be taken a serious side effect of software engineering. This paper suggests a solution by building up a library of solvers using spreadsheets, with the effect that the encapsulated knowledge of building modelling solvers can permanently be brought to life in education with the active learning culture. Implementation was carried out in the same way for steady state flow as well as explicit 2D and 3D finite difference approximation for transient flow. This study raises concern about the encapsulated body of knowledge contributed to the emergence and the establishment of modelling software applications since 1980. This body of knowledge comprise a deeper understanding of equations of often partial differential equations describing physical problems, as well as their numerical transformation into systems of equations and their subsequent properly- and improperly posed systems of equation in terms of their assumptions and quality conditions. The outcome is the emergence of a cookbook mentality among the new breed of mathematical modelers without any critical thinking. The results revealed that spreadsheet can be used with the aid of the Solver function. This idea capitalized on the capabilities of the net-generation and opens up the possibility for the emergence of bottom-up open source modelling platforms.

Cite This Paper

Farzin Salmasi, Mohammad Taghi Sattari, Halit Apaydin, " Mathematical Based Implicit and Explicit Finite Difference Techniques for Solving the Ground Water Flow Equations Using Spreadsheets", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.8, No.4, pp. 1-14, 2022. DOI: 10.5815/ijmsc.2022.04.01


[1]Scriven, M., Paul, R., 2003. Defining Critical Thinking: A draft statement prepared for the National Council for Excellence in Critical Thinking Instruction. 

[2]Kyrpychenko O, Pushchyna I, Kichuk Y, Shevchenko N, Luchaninova O, Koval V, 2021. Communicative Competence Development in Teaching Professional Discourse in Educational Establishments, International Journal of Modern Education and Computer Science(IJMECS), Vol.13, No.4, pp. 16-27, DOI: 10.5815/ijmecs.2021.04.02

[3]Jewell, T.K., 2001. Teaching hydraulic design using equation solvers. J. Hydraul. Eng. 127(12), 1013–1021.

[4]Cheetancheri, K.G. and Cheng H.H. 2009. Spreadsheet-based interactive design and analysis of mechanisms using Excel and Ch, Adv Eng Softw 40, 274-280.  doi:10.1016/j.advengsoft.2007.08.003.

[5]Yudianto MRA, Agustin T, James RM, Rahma FI, Rahim A, Utami E, 2021. Rainfall Forecasting to Recommend Crops Varieties Using Moving Average and Naive Bayes Methods, International Journal of Modern Education and Computer Science(IJMECS), 13 (3), 23-33, DOI: 10.5815/ijmecs.2021.03.03

[6]El-Hajj, A., Kabalan, K.Y., Al-Husseini, M., 2003. Antenna array design using spreadsheets. IEEE Trans Edu 46(3), 319–24. DOI: 10.1109/TE.2003.813518.

[7]Shoup, T.E., 2004. Using spreadsheet modules to augment tolerance dimensioning. In: Proceedings of the DETC. 2912, ASME, 36–44. DOI: 10.1115/DETC2004-57649.

[8]Karahan, H. 2006. Implicit finite difference techniques for the advection–diffusion equation using spreadsheets. Adv Eng Softw 37, 601-608.

[9]Dehghan, M., & Mohammadi, V. (2017). Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model. Communications in Nonlinear Science and Numerical Simulation, 44, 204–219.

[10]Panigrahi, P. K., & Velusamy, K. (2019). A robust multiphase model to investigate molten material relocation during total flow blockage in SFR fuel subassembly. Annals of Nuclear Energy, 133, 59–72.

[11]Macías-Díaz, J. E., & Villa-Morales, J. (2017). A deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution. Journal of Computational and Applied Mathematics, 318, 93–106.

[12]Sattari MT., Mirabbasi R., Sushab RS, Abraham J. 2018. Prediction of Groundwater Level in Ardebil Plain Using Support Vector Regression and M5 Tree Model. Groundwater 56(4):636-646.

[13]Cryer, C.W., 1970. On the approximate solution of free boundary problems using finite differences. Journal of the Association for Computing Machinery 17(3), 397–411. doi:10.1145/321592.321593.

[14]Weiss, P.T., Gulliver, J.S., 2001. What do students need in hydraulic design projects? J. Hydraul. Eng. 127(12), 984–991. DOI: 10.1061/(ASCE)0733-9429(2001)127:12(984).

[15]Huddleston, D.H., 2002. Spreadsheet tools utilised to introduce computational field simulation concepts to undergraduate engineering students. Comput. Educ. J. 12(1), 6–11.

[16]Huddlesston, D.H., Alarcon Vladimir J., Chen, W., 2004. Water distribution network analysis using Excel. Journal of hydraulic engineering 103(10), 1033-1036.

[17]Pandit A. (2016). Water engineering with the spreadsheet: A workbook for water resources calculations using excel, Publisher: Reston, Virginia: American Society of Civil Engineers, DOI: 10.1061/9780784414040.

[18]Niazkar M. and Afzali S.H. (2015). Application of Excel spreadsheet in engineering education. First International & Fourth National Conference on Engineering Education Shiraz University, 10-12 November 2015. Pp 1-7.

[19]Divayana DGH, Ariawan IPW, Giri MKW, CIPP-SAW, 2021. Application as an Evaluation Tool of E-Learning Effectiveness, International Journal of Modern Education and Computer Science (IJMECS), 13 (6), 42-59, DOI: 10.5815/ijmecs.2021.06

[20]Salmasi, F., Azamathulla, H. M., (2013). Determination of optimum relaxation coefficient using finite difference method for groundwater flow, Arabian Journal of Geo-sciences, (6): 3409-3415,

[21]Bear, J., 1979. Hydraulics of Groundwater, Dover Books on Engineering.

[22]Hosseinzadeh Asl, R., Salmasi, F., Arvanaghi, H., (2020). Numerical investigation on geometric configurations affecting seepage from unlined earthen channels and the comparison with field measurements. Engineering Applications of Computational Fluid Mechanics, 14:1, 236-253,

[23]Wang, H.F., Anderson, M.P., 1982. Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods. Academic Press, NY, ISBN-13: 978-0127345857, pp: 237.

[24]Lindstrom, R., 1994. The Business Week Guide to Multimedia Presentations: Create Dynamic Presentations That Inspire, New York: McGraw-Hill.

[25]Prensky, M., 2007. Digital Game-Based Learning (Kindle Edition), Paragon House Publishing.

[26]Brandes, D., Ginnis, G., 1994. A Guide to Student-Centred Learning.