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A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme

Year 2018, Volume: 20 Issue: 3, 36 - 52, 29.10.2018
https://doi.org/10.25092/baunfbed.476583

Abstract

This paper describes a numerical solution for the advection-diffusion equation.  The proposed method is based on the operator splitting method which helps to obtain accurate solutions.  That is, instead of sum, the operators are considered separately for the physical compatibility.  In the process, method of characteristics combined with cubic spline interpolation and Saulyev method are used in sub-operators, respectively.  After guaranteeing the convergence of the method the efficiency is also tested on one-dimensional advection-diffusion problem for a wide range of Courant numbers which plays a crucial role on the convergence of the solution.  The obtained results are compared with the analytical solution of the problem and other solutions which are available in the literature.  It is revealed that the proposed method produces good approach not only for small Caurant numbers but also big ones even though it is explicit method.

References

  • Srivastava, R., Flow through open channels, Oxford University Press, (2008).
  • Appadu, A. R., Numerical solution of the 1D advection-diffusion equation using standard and nonstandard finite difference schemes, Journal of Applied Mathematics, 2013, 1-14, (2013).
  • Price, H. S., Cavendish, J. C. and Varga, R. S., Numerical methods of higher-order accuracy for diffusion-convection equations, Society of Petroleum Engineers, 8, 3, 293-303, (1968).
  • Gurarslan, G., Karahan, H., Alkaya, D., Sari, M. and Yasar M., Numerical solution of advection-diffusion equation using a sixth-order compact finite difference method, Mathematical Problems in Engineering, 2013, 1-7, (2013).
  • Gurarslan, G., Accurate simulation of contaminant transport using high-order compact finite difference schemes, Journal of Applied Mathematics, 2014, 1-8, (2014).
  • Taigbenu, A. E. and Onyejekwe, O. O., Transient 1D transport equation simulated by a mixed green element formulation, International Journal for Numerical Methods in Fluids, 25, 4, 437-454, (1997).
  • Mittal, R. C. and Jain, R. K., Numerical solution of convection-diffusion equation using cubic B-splines collocation methods with Neumann’s boundary conditions, International Journal of Applied Mathematics and Computation, 4, 2, 115-127, (2012).
  • Goh, J., Majid, A. A. and Ismail, A. I. M., Cubic B-spline collocation method for one-dimensional heat and advection-diffusion equations, Journal of Applied Mathematics, 2012, 1-8, (2012).
  • Irk, D., Dag, I. and Tombul, M., Extended cubic B-spline solution of the advection-diffusion equation, KSCE Journal of Civil Engineering, 19, 4, 929-934, (2015).
  • Korkmaz, A. and Dag, I., Cubic B-spline differential quadrature methods for the advection-diffusion equation, International Journal of Numerical Methods for Heat and Fluid Flow, 22, 8, 1021-1036, (2012).
  • Korkmaz, A. and Dag, I., Quartic and quintic B-spline collocation methods for advection-diffusion equation, Applied Mathematics and Computation, 274, 208-219, (2016).
  • Holly Jr., F. M. and Preissmann, A., Accurate calculation of transport in two dimensions, Journal of Hydraulic Division, 103, 11, 1259-1277, (1977).
  • Tsai, T.-L., Yang, J.-C. and Huang, L.-H., Characteristics method using cubic-spline interpolation for advection-diffusion equation, Journal of Hydraulic Engineering, 130, 6, 580-585, (2004).
  • Tsai, T.-L., Chiang, S.-W. and Yang, J.-C., Examination of characteristics method with cubic interpolation for advection-diffusion equation, Computers and Fluids, 35, 10, 1217-1227, (2006).
  • Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A. and Rashid, A., The numerical solution of advection-diffusion problems using new cubic trigonometric B-spline approach, Applied Mathematical Modelling, 40, 4586-4611, (2016).
  • Dag, I., Canivar, A. and Sahin, A., Taylor-Galerkin method for advection-diffusion equation, Kybernetes, 40, 5/6, 762-777, (2011).
  • Zhou, J. G., A lattice Boltzmann method for solute transport, International Journal for Numerical Methods in Fluids, 61, 848-863, (2009).
  • Bak, S., Bu, S. and Kim, P., An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-diffusion Equation, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, 8, 8, (2014).
  • Hundsdorfer, W. and Verwer, J., Numerical solution of time-dependent advection-diffusion-reaction equations, Springer-Verlag Berlin Heidelberg, (2003).
  • Bahar, E., Numerical solution of advection-dispersion equation using operator splitting method, MSc Thesis, Pamukkale University, (in Turkish) (2017).
  • Bahar, E., and Gurarslan, G., Numerical solution of advection-diffusion equation using operator splitting method, International Journal of Engineering and Applied Sciences, 9, 4, 76-88, (2017).
  • Osterby O., Bieniasz L. K. and Britz D., Numerical stability of the Saul’yev Finite Difference Algorithms for Electrochemical Kinetic Simulation: Matrix Stability Analysis for an Example Problem Involving Mixed Boundary Conditions, Computers and Chemistry, 19, 4, 357-370, (1995).
  • 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).
  • Sankaranarayanan, S., Shankar, N. J. and Cheong, H. F., Three-dimensional finite difference model for transport of conservative pollutants, Ocean Engineering, 25, 6, 425-442, (1998).
  • Gardner, L.R.T. and Dag, I., A numerical solution of the advection-diffusion equation using b-spline finite element, Proceedings International AMSE Conference, 109-116, Lyon, France, (1994).
  • Dag, I., Irk, D. and Tombul, M., Least-squares finite element method for the advection-diffusion equation, Applied Mathematics and Computation, 173, 1, 554–565, (2006).

Yarı-Lagrangian bir şema yaklaşımına dayalı adveksiyon-difüzyon denkleminin bir sayısal çözümü

Year 2018, Volume: 20 Issue: 3, 36 - 52, 29.10.2018
https://doi.org/10.25092/baunfbed.476583

Abstract

Bu çalışmada, adveksiyon-difüzyon denklemi için sayısal bir çözüm tanıtılmaktadır. Önerilen yöntem, doğru çözümler elde edilmesine yardımcı olan operatör ayırma metoduna dayanmaktadır.  Yani, toplam yerine, operatörler fiziksel uyumluluk için ayrı olarak ele alınmaktadır.  Bu süreçte, alt operatörler için karakteristikler yöntemi ile bir araya getirilmiş kübik spline interpolasyonu ve Saulyev metodu sırası ile kullanılmıştır.  Yöntemin yakınsamasını garanti altına aldıktan sonra, verimlilik de çözümün yakınsaması üzerinde önemli bir rol oynayan farklı Courant sayıları için tek boyutlu adveksiyon-difüzyon problemi üzerinde test edilmiştir.  Elde edilen sonuçlar, problemin analitik çözümü ve literatürde mevcut olan diğer çözümlerle karşılaştırılmıştır.  Önerilen yöntemin, açık bir yöntem olmasına rağmen sadece küçük Caurant sayıları için değil, büyük olanlar için de iyi bir yaklaşım oluşturduğu ortaya çıkmıştır.

References

  • Srivastava, R., Flow through open channels, Oxford University Press, (2008).
  • Appadu, A. R., Numerical solution of the 1D advection-diffusion equation using standard and nonstandard finite difference schemes, Journal of Applied Mathematics, 2013, 1-14, (2013).
  • Price, H. S., Cavendish, J. C. and Varga, R. S., Numerical methods of higher-order accuracy for diffusion-convection equations, Society of Petroleum Engineers, 8, 3, 293-303, (1968).
  • Gurarslan, G., Karahan, H., Alkaya, D., Sari, M. and Yasar M., Numerical solution of advection-diffusion equation using a sixth-order compact finite difference method, Mathematical Problems in Engineering, 2013, 1-7, (2013).
  • Gurarslan, G., Accurate simulation of contaminant transport using high-order compact finite difference schemes, Journal of Applied Mathematics, 2014, 1-8, (2014).
  • Taigbenu, A. E. and Onyejekwe, O. O., Transient 1D transport equation simulated by a mixed green element formulation, International Journal for Numerical Methods in Fluids, 25, 4, 437-454, (1997).
  • Mittal, R. C. and Jain, R. K., Numerical solution of convection-diffusion equation using cubic B-splines collocation methods with Neumann’s boundary conditions, International Journal of Applied Mathematics and Computation, 4, 2, 115-127, (2012).
  • Goh, J., Majid, A. A. and Ismail, A. I. M., Cubic B-spline collocation method for one-dimensional heat and advection-diffusion equations, Journal of Applied Mathematics, 2012, 1-8, (2012).
  • Irk, D., Dag, I. and Tombul, M., Extended cubic B-spline solution of the advection-diffusion equation, KSCE Journal of Civil Engineering, 19, 4, 929-934, (2015).
  • Korkmaz, A. and Dag, I., Cubic B-spline differential quadrature methods for the advection-diffusion equation, International Journal of Numerical Methods for Heat and Fluid Flow, 22, 8, 1021-1036, (2012).
  • Korkmaz, A. and Dag, I., Quartic and quintic B-spline collocation methods for advection-diffusion equation, Applied Mathematics and Computation, 274, 208-219, (2016).
  • Holly Jr., F. M. and Preissmann, A., Accurate calculation of transport in two dimensions, Journal of Hydraulic Division, 103, 11, 1259-1277, (1977).
  • Tsai, T.-L., Yang, J.-C. and Huang, L.-H., Characteristics method using cubic-spline interpolation for advection-diffusion equation, Journal of Hydraulic Engineering, 130, 6, 580-585, (2004).
  • Tsai, T.-L., Chiang, S.-W. and Yang, J.-C., Examination of characteristics method with cubic interpolation for advection-diffusion equation, Computers and Fluids, 35, 10, 1217-1227, (2006).
  • Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A. and Rashid, A., The numerical solution of advection-diffusion problems using new cubic trigonometric B-spline approach, Applied Mathematical Modelling, 40, 4586-4611, (2016).
  • Dag, I., Canivar, A. and Sahin, A., Taylor-Galerkin method for advection-diffusion equation, Kybernetes, 40, 5/6, 762-777, (2011).
  • Zhou, J. G., A lattice Boltzmann method for solute transport, International Journal for Numerical Methods in Fluids, 61, 848-863, (2009).
  • Bak, S., Bu, S. and Kim, P., An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-diffusion Equation, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, 8, 8, (2014).
  • Hundsdorfer, W. and Verwer, J., Numerical solution of time-dependent advection-diffusion-reaction equations, Springer-Verlag Berlin Heidelberg, (2003).
  • Bahar, E., Numerical solution of advection-dispersion equation using operator splitting method, MSc Thesis, Pamukkale University, (in Turkish) (2017).
  • Bahar, E., and Gurarslan, G., Numerical solution of advection-diffusion equation using operator splitting method, International Journal of Engineering and Applied Sciences, 9, 4, 76-88, (2017).
  • Osterby O., Bieniasz L. K. and Britz D., Numerical stability of the Saul’yev Finite Difference Algorithms for Electrochemical Kinetic Simulation: Matrix Stability Analysis for an Example Problem Involving Mixed Boundary Conditions, Computers and Chemistry, 19, 4, 357-370, (1995).
  • 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).
  • Sankaranarayanan, S., Shankar, N. J. and Cheong, H. F., Three-dimensional finite difference model for transport of conservative pollutants, Ocean Engineering, 25, 6, 425-442, (1998).
  • Gardner, L.R.T. and Dag, I., A numerical solution of the advection-diffusion equation using b-spline finite element, Proceedings International AMSE Conference, 109-116, Lyon, France, (1994).
  • Dag, I., Irk, D. and Tombul, M., Least-squares finite element method for the advection-diffusion equation, Applied Mathematics and Computation, 173, 1, 554–565, (2006).
There are 26 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Ersin Bahar 0000-0001-8892-8441

Sıla O. Korkut 0000-0003-4784-2013

Yesim Cıcek This is me 0000-0001-5438-4685

Gurhan Gurarslan 0000-0002-9796-3334

Publication Date October 29, 2018
Submission Date August 18, 2018
Published in Issue Year 2018 Volume: 20 Issue: 3

Cite

APA Bahar, E., O. Korkut, S., Cıcek, Y., Gurarslan, G. (2018). A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(3), 36-52. https://doi.org/10.25092/baunfbed.476583
AMA Bahar E, O. Korkut S, Cıcek Y, Gurarslan G. A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme. BAUN Fen. Bil. Enst. Dergisi. October 2018;20(3):36-52. doi:10.25092/baunfbed.476583
Chicago Bahar, Ersin, Sıla O. Korkut, Yesim Cıcek, and Gurhan Gurarslan. “A Numerical Solution for Advection-Diffusion Equation Based on a Semi-Lagrangian Scheme”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20, no. 3 (October 2018): 36-52. https://doi.org/10.25092/baunfbed.476583.
EndNote Bahar E, O. Korkut S, Cıcek Y, Gurarslan G (October 1, 2018) A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20 3 36–52.
IEEE E. Bahar, S. O. Korkut, Y. Cıcek, and G. Gurarslan, “A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme”, BAUN Fen. Bil. Enst. Dergisi, vol. 20, no. 3, pp. 36–52, 2018, doi: 10.25092/baunfbed.476583.
ISNAD Bahar, Ersin et al. “A Numerical Solution for Advection-Diffusion Equation Based on a Semi-Lagrangian Scheme”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20/3 (October 2018), 36-52. https://doi.org/10.25092/baunfbed.476583.
JAMA Bahar E, O. Korkut S, Cıcek Y, Gurarslan G. A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme. BAUN Fen. Bil. Enst. Dergisi. 2018;20:36–52.
MLA Bahar, Ersin et al. “A Numerical Solution for Advection-Diffusion Equation Based on a Semi-Lagrangian Scheme”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 20, no. 3, 2018, pp. 36-52, doi:10.25092/baunfbed.476583.
Vancouver Bahar E, O. Korkut S, Cıcek Y, Gurarslan G. A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme. BAUN Fen. Bil. Enst. Dergisi. 2018;20(3):36-52.