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The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method

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Abstract

A fourth-order numerical method based on cubic B-spline functions has been proposed to solve a class of advection–diffusion equations. The proposed method has several advantageous features such as high accuracy and fast results with very small CPU time. We have applied the Crank–Nicolson method to solve the advection–diffusion equation. The stability analysis is performed, and the method is shown to be unconditionally stable. Error analysis is carried out to show that the proposed method has fourth-order convergence. The efficiency of the proposed B-spline method has been checked by applying on ten important advection–diffusion problems of three types, having Dirichlet, Neumann and periodic boundary conditions. Considered examples prove the mentioned advantages of the method. The computed results are also compared with those available in the literature, and it is found that our method is giving better results.

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References

  1. Parlarge, J.Y.: Water transport in soils. Ann. Rev. Fluid Mech. 2, 77–102 (1980)

    Article  Google Scholar 

  2. Zlatev, Z., Berkowicz, R., Prahm, L.P.: Implementation of a variable step-size variable formula in the time-integration part of a code for treatment of longrange transport of air pollutants. J. Comput. Phys. 55 (1984)

  3. Chatwin, P.C., Allen, C.M.: Mathematical models of dispersion in rivers and estuaries. Ann. Rev. Fluid Mech. 17, 119–149 (1985)

    Article  Google Scholar 

  4. Mittal, R.C., Jain, R.K.: Redefined cubic B-spline collocation method for solving convection diffusion equations. Appl. Math. Model. 36, 5555–5573 (2012)

    Article  MathSciNet  Google Scholar 

  5. Dehghan, M.: Weighted finite difference techniques for the one-dimensional advection–diffusion equation. Appl. Math. Comput. 147, 307–319 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Rizwan-Uddin: A second-order space and time nodal method for the one-dimensional convection–diffusion equation. Comput. Fluids 26, 233–247 (1997)

    Article  MathSciNet  Google Scholar 

  7. Mohebbi, A., Dehghan, M.: High-order compact solution of the one-dimensional heat and advection–diffusion equations. Appl. Math. Model. 34, 3071–3084 (2010)

    Article  MathSciNet  Google Scholar 

  8. Dehghan, M.: Numerical solution of the three-dimensional advection–diffusion equation. Appl. Math. Comput. 150, 5–19 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Dehghan, M., Mohebbi, A.: High-order compact boundary value method for the solution of unsteady convection–diffusion problems. Math. Comput. Simul. 79, 683–699 (2010)

    Article  MathSciNet  Google Scholar 

  10. Dehghan, M.: Quasi-implicit and two-level explicit finite-difference procedures for solving the one-dimensional advection equation. Appl. Math.Comput. Simul. 167, 46–67 (2005)

    Article  MathSciNet  Google Scholar 

  11. Singh, I., Kumar, S.: Approximate solution of convection diffusion equation using Haar wavelet. Italian J. Pure Appl. Math. 35, 143–154 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Salkuyeh, D.K.: On the finite difference approximation to the convection-diffusion equation. Appl. Math. Comput. 179 (2006)

  13. Karahan, H.: Unconditional stable explicit finite difference technique for the advection–diffusion equation using spreadsheets. Adv. Eng. Softw. 38, 80–86 (2007)

    Article  Google Scholar 

  14. Karahan, H.: Implicit finite difference techniques for the advection–diffusion equation using spreadsheets. Adv. Eng. Softw. 37, 601–608 (2006)

    Article  Google Scholar 

  15. Ding, H.F., Zhang, Y.X.: A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations. J. Comput. Appl. Math. 230, 600–606 (2009)

    Article  MathSciNet  Google Scholar 

  16. Szymkiewicz, R.: Solution of the advection–diffusion equation using the spline function and finite elements. Commun. Numer. Methods Eng. 9, 197–206 (1993)

    Article  MathSciNet  Google Scholar 

  17. Mohammadi, R.: Exponential B-spline solution of convection–diffusion equations. Appl. Math. 4, 933–944 (2013)

    Article  Google Scholar 

  18. Mittal, R.C., Jain, R.K.: Numerical solution of convection–diffusion equation using cubic B-splines collocation methods with Neumann’s boundary conditions. Int. J. Appl. Math. Comput. 4, 115–127 (2012)

    Google Scholar 

  19. Goh, J., Majid, A.A., Ismail, A.I.B. Md.: Cubic B-spline collocation method for one-dimensional heat and advection–diffusion equations. J. Appl. Math. Article ID 458701 (2012)

  20. Korkmaz, A., Dag, I.: Cubic B-spline differential quadrature methods for the advection–diffusion equation. Int. J. Numer. Methods Heat Fluid Flow 22, 1021–1036 (2012)

    Article  MathSciNet  Google Scholar 

  21. Gardner, L.R.T., Dag, I.: A numerical solution of the advection-diffusion equation using B-spline finite element. In: Proceedings of the International AMSE Conference on Systems Analysis, Control and Design, Lyon, France, pp. 109–116 (1994)

  22. Dag, I., Canivar, A., Sahin, A.: Taylor–Galerkin method for advection–diffusion equation. Kybernetes 40, 762–777 (2011)

    Article  MathSciNet  Google Scholar 

  23. Kadalbajoo, M.K., Arora, P.: Taylor–Galerkin B-spline finite element method for the one dimensional advection–diffusion equation. Numer. Methods Par. Diff. Eqs. 26, 1206–1223 (2009)

    Article  MathSciNet  Google Scholar 

  24. Prenter, P.M.: Splines and Variational Methods. Wiley, Hoboken (1989)

    MATH  Google Scholar 

  25. Lucas, T.R.: Error bounds for interpolating cubic splines under various end conditions. Siam J. Numer. Anal. 11, 569–584 (1975)

    Article  MathSciNet  Google Scholar 

  26. Sastry, S.S.: Introductory Methods of Numerical Analysis, fourth ed., PHI Learning (2009)

  27. Salama, A.A., Zidan, H.Z.: Fourth order schemes of exponential type for singularly perturbed parabolic partial differential equations. J. Math. 36 (2006)

  28. Nazir, T., Abbas, M., Ahmad Izani, M., Ismail, A.A., Majid, A.R.: The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach. Appl. Math. Model. 40, 4586–4611 (2016)

    Article  MathSciNet  Google Scholar 

  29. Chawla, M.M., Al-Zanaidi, M.A., Al-Aslab, M.G.: Extended one step time-integration schemes for convection–diffusion equations. Comput. Math. Appl. 39, 71–84 (2000)

    Article  MathSciNet  Google Scholar 

  30. Korkmaz, A., Dag, I.: Quartic and quintic B-spline methods for advection–diffusion equation. Appl. Math. Comput. 274, 208–219 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Cao, H.-H., Liu, L.-B., Zhang, Y., Fu, S.: A fourth-order method of the convection–diffusion equations with Neumann boundary conditions. Appl. Math. Comput. 217, 9133–9141 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Chen, N., Haiming, G.: Alternating group explicit iterative methods for one-dimensional advection–diffusion equation. Am. J. Comput. Math. 5, 274–282 (2015)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, 247667, India

    R. C. Mittal & Rajni Rohila

  2. Department of Applied Sciences, The NorthCap University, Gurugram, Haryana, India

    R. C. Mittal & Rajni Rohila

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  1. R. C. Mittal
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  2. Rajni Rohila
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Correspondence to Rajni Rohila.

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Mittal, R.C., Rohila, R. The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method. Math Sci 14, 409–423 (2020). https://doi.org/10.1007/s40096-020-00352-7

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  • DOI: https://doi.org/10.1007/s40096-020-00352-7

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