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Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method

Year 2017, Volume: 9 Issue: 4, 76 - 88, 27.12.2017

Ersin Bahar Gürhan Gürarslan

https://doi.org/10.24107/ijeas.357237
Cited By: 7

Abstract

In this study, effects of operator
splitting methods to the solution of advection-diffusion equation are examined.
Within the context of this work two operator splitting methods, Lie-Trotter and
Strang splitting methods were used and comparisons were made through various
Courant numbers. These methods have been implemented to advection-diffusion equation
in one-dimension. Numerical solutions of advection and dispersion processes
were carried out by a characteristics method with cubic spline interpolation
(MOC-CS) and Crank-Nicolson (CN) finite difference scheme, respectively.
Obtained results were compared with analytical solutions of the problems and
available methods in the literature. It is seen that MOC-CS-CN method has lower
error norm values than the other methods. MOC-CS-CN produces accurate results
even while the time steps are great.

Keywords

Advection-diffusion equation Operator splitting methods Method of characteristics Finite difference

References

  • [1] Srivastava, R., Flow Through Open Channels, Oxford University Press, 2008.
  • [2] Baptista, A. E. De M., Solution of Advection-Dominated Transport by Eulerian-Lagrangian Methods Using the Backward Method of Characteristics, Ph.D Thesis, MIT, Cambridge, 1987.
  • [3] Holly, F.M, Usseglio‐Polatera, J., Dispersion Simulation in Two‐dimensional Tidal Flow. Journal of Hydraulic Engineering, 110(7), 905–926, 1984.
  • [4] Chen, Y., Falconer, R.A., Advection-diffusion modelling using the modified QUICK scheme, International Journal for Numerical Methods in Fluids, 15(10), 1171–1196, 1992.
  • [5] Szymkiewicz, R., Solution of the advection-diffusion equation using the spline function and finite elements, Communications in Numerical Methods in Engineering, 9, 197–206, 1993.
  • [6] Ahmad, Z., Kothyari, U.C., Time-line cubic spline interpolation scheme for solution of advection equation, Computers and Fluids, 30(6), 737–752, 2001.
  • [7] Tsai, T.L., Yang, J.C., Huang, L.H., Characteristics Method Using Cubic–Spline Interpolation for Advection–Diffusion Equation, Journal of Hydraulic Engineering, 130(6), 580–585, 2004.
  • [8] Verma, P., Prasad, K.S.H., Ojha, C.S.P., MacCormack scheme based numerical solution of advection-dispersion equation, ISH Journal of Hydraulic Engineering, 12(1), 27-38, 2006.
  • [9] Tian, Z.F., Ge, Y.B., A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, Journal of Computational and Applied Mathematics, 198(1), 268–286, 2007.
  • [10] Sari, M., Gürarslan, G., Zeytinoǧlu, A., High-order finite difference schemes for solving the advection-diffusion equation, Mathematical and Computational Applications, 15(3), 449–460, 2010.
  • [11] Gurarslan, G., Karahan, H., Alkaya, D., Sari, M., Yasar, M., Numerical solution of advection-diffusion equation using a sixth-order compact finite difference method, Mathematical Problems in Engineering, vol. 2013, Article ID 672936, 7 pages, 2013.
  • [12] Gurarslan, G., Accurate Simulation of Contaminant Transport Using High-Order Compact Finite Difference Schemes, Journal of Applied Mathematics, 2014, 1–8, 2014.
  • [13] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag Berlin Heidelberg, 2003.
  • [14] Esfandiari, R.S., Numerical Methods for Engineers and Scientists Using MATLAB, New York: Taylor and Francis Group, 2013.
  • [15] Lam, C.Y., Applied Numerical Methods for Partial Differential Equations, Singapore, Simon and Schuster, 1994.
  • [16] Nazir, T., Abbas, M., Izani, A., Abd, A., The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach, Applied Mathematical Modelling, 40(7–8), 4586–4611, 2016.
  • [17] Irk, D., Dag, I., Tombul, M., Extended Cubic B-spline Solution of the Advection-Diffusion Equation, KSCE Journal of Civil Engineering, 19(4), 929-934, 2015.
  • [18] Holly, F.M., Preissmann, A., Accurate calculation of transport in two dimensions, Journal of Hydraulic Division, 103(11), 1259-1277, 1977.
  • [19] Gardner, L.R.T. and Dag, I., A numerical solution of the advection-diffusion equation using B-spline finite element, Proceedings International AMSE Conference, Lyon, France, 109-116, 1994.
  • [20] Dag, I., Irk, D., Tombul, M., Least-squares finite element method for the advection-diffusion equation, Applied Mathematics and Computation, 173(1), 554–565, 2006.
  • [21] Dag, I., Canivar, A., Sahin, A., Taylor‐Galerkin method for advection‐diffusion equation, Kybernetes, 40(5/6), 762–777, 2011.
  • [22] Bahar, E., Numerical Solution of Advection-Dispersion Equation Using Operator Splitting Method, Denizli: Pamukkale University, (in Turkish) 2017.

Year 2017, Volume: 9 Issue: 4, 76 - 88, 27.12.2017

Ersin Bahar Gürhan Gürarslan

https://doi.org/10.24107/ijeas.357237
Cited By: 7

Abstract

References

  • [1] Srivastava, R., Flow Through Open Channels, Oxford University Press, 2008.
  • [2] Baptista, A. E. De M., Solution of Advection-Dominated Transport by Eulerian-Lagrangian Methods Using the Backward Method of Characteristics, Ph.D Thesis, MIT, Cambridge, 1987.
  • [3] Holly, F.M, Usseglio‐Polatera, J., Dispersion Simulation in Two‐dimensional Tidal Flow. Journal of Hydraulic Engineering, 110(7), 905–926, 1984.
  • [4] Chen, Y., Falconer, R.A., Advection-diffusion modelling using the modified QUICK scheme, International Journal for Numerical Methods in Fluids, 15(10), 1171–1196, 1992.
  • [5] Szymkiewicz, R., Solution of the advection-diffusion equation using the spline function and finite elements, Communications in Numerical Methods in Engineering, 9, 197–206, 1993.
  • [6] Ahmad, Z., Kothyari, U.C., Time-line cubic spline interpolation scheme for solution of advection equation, Computers and Fluids, 30(6), 737–752, 2001.
  • [7] Tsai, T.L., Yang, J.C., Huang, L.H., Characteristics Method Using Cubic–Spline Interpolation for Advection–Diffusion Equation, Journal of Hydraulic Engineering, 130(6), 580–585, 2004.
  • [8] Verma, P., Prasad, K.S.H., Ojha, C.S.P., MacCormack scheme based numerical solution of advection-dispersion equation, ISH Journal of Hydraulic Engineering, 12(1), 27-38, 2006.
  • [9] Tian, Z.F., Ge, Y.B., A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, Journal of Computational and Applied Mathematics, 198(1), 268–286, 2007.
  • [10] Sari, M., Gürarslan, G., Zeytinoǧlu, A., High-order finite difference schemes for solving the advection-diffusion equation, Mathematical and Computational Applications, 15(3), 449–460, 2010.
  • [11] Gurarslan, G., Karahan, H., Alkaya, D., Sari, M., Yasar, M., Numerical solution of advection-diffusion equation using a sixth-order compact finite difference method, Mathematical Problems in Engineering, vol. 2013, Article ID 672936, 7 pages, 2013.
  • [12] Gurarslan, G., Accurate Simulation of Contaminant Transport Using High-Order Compact Finite Difference Schemes, Journal of Applied Mathematics, 2014, 1–8, 2014.
  • [13] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag Berlin Heidelberg, 2003.
  • [14] Esfandiari, R.S., Numerical Methods for Engineers and Scientists Using MATLAB, New York: Taylor and Francis Group, 2013.
  • [15] Lam, C.Y., Applied Numerical Methods for Partial Differential Equations, Singapore, Simon and Schuster, 1994.
  • [16] Nazir, T., Abbas, M., Izani, A., Abd, A., The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach, Applied Mathematical Modelling, 40(7–8), 4586–4611, 2016.
  • [17] Irk, D., Dag, I., Tombul, M., Extended Cubic B-spline Solution of the Advection-Diffusion Equation, KSCE Journal of Civil Engineering, 19(4), 929-934, 2015.
  • [18] Holly, F.M., Preissmann, A., Accurate calculation of transport in two dimensions, Journal of Hydraulic Division, 103(11), 1259-1277, 1977.
  • [19] Gardner, L.R.T. and Dag, I., A numerical solution of the advection-diffusion equation using B-spline finite element, Proceedings International AMSE Conference, Lyon, France, 109-116, 1994.
  • [20] Dag, I., Irk, D., Tombul, M., Least-squares finite element method for the advection-diffusion equation, Applied Mathematics and Computation, 173(1), 554–565, 2006.
  • [21] Dag, I., Canivar, A., Sahin, A., Taylor‐Galerkin method for advection‐diffusion equation, Kybernetes, 40(5/6), 762–777, 2011.
  • [22] Bahar, E., Numerical Solution of Advection-Dispersion Equation Using Operator Splitting Method, Denizli: Pamukkale University, (in Turkish) 2017.
There are 22 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Ersin Bahar

Gürhan Gürarslan

Publication Date December 27, 2017
Acceptance Date December 6, 2017
Published in Issue Year 2017 Volume: 9 Issue: 4

Cite

APA Bahar, E., & Gürarslan, G. (2017). Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method. International Journal of Engineering and Applied Sciences, 9(4), 76-88. https://doi.org/10.24107/ijeas.357237
AMA Bahar E, Gürarslan G. Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method. IJEAS. December 2017;9(4):76-88. doi:10.24107/ijeas.357237
Chicago Bahar, Ersin, and Gürhan Gürarslan. “Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method”. International Journal of Engineering and Applied Sciences 9, no. 4 (December 2017): 76-88. https://doi.org/10.24107/ijeas.357237.
EndNote Bahar E, Gürarslan G (December 1, 2017) Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method. International Journal of Engineering and Applied Sciences 9 4 76–88.
IEEE E. Bahar and G. Gürarslan, “Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method”, IJEAS, vol. 9, no. 4, pp. 76–88, 2017, doi: 10.24107/ijeas.357237.
ISNAD Bahar, Ersin - Gürarslan, Gürhan. “Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method”. International Journal of Engineering and Applied Sciences 9/4 (December 2017), 76-88. https://doi.org/10.24107/ijeas.357237.
JAMA Bahar E, Gürarslan G. Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method. IJEAS. 2017;9:76–88.
MLA Bahar, Ersin and Gürhan Gürarslan. “Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method”. International Journal of Engineering and Applied Sciences, vol. 9, no. 4, 2017, pp. 76-88, doi:10.24107/ijeas.357237.
Vancouver Bahar E, Gürarslan G. Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method. IJEAS. 2017;9(4):76-88.

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