Numerical Solutions of the 1D Convection–Diffusion–Reaction and the Burgers Equation Using Implicit Multi-stage and Finite Element Methods



In this work we apply the semi-discrete formulation, where the time variable is discretized using an implicit multi-stage method and the space variable is discretized using the finite element method, to obtain numerical solutions for the 1D convection–diffusion–reaction and the Burgers equation, whose analytical solutions are known. More specifically, we use the implicit multi-stage method of second and fourth-order for time discretization. For space discretization, we use three finite elements methods, least square (LSFEM), Galerkin (GFEM) and streamline-upwind Petrov-Galerkin (SUPG). We present an error analysis, comparing the numerical with analytical solutions. We verify that the implicit multi-stage second-order method when combined with the LSFEM, GFEM and SUPG, increased the region of convergence of the numerical solutions. LSFEM presented the better performance when compared to GFEM and SUPG.


1D convection–diffusion–reaction equation Burgers equation Implicit multi-stage methods Finite elements methods 


  1. [BeNa08]
    Behmardi, D., Nayeri, D.E.: Introduction of Fréchet and Gâteaux derivative. Appl. Math. Sci. 2, 975–980 (2008)MathSciNetMATHGoogle Scholar
  2. [DoRoHu00]
    Donea, J., Roig, B., Huerta, A.: Higher-order accurate time-stepping schemes for convection-difusion problems. Comput. Meth. Appl. Mech. Eng. 182, 249–275 (2000)MathSciNetMATHCrossRefGoogle Scholar
  3. [DoRpHu03]
    Donea, J., Roig, B., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Chichester (2003)CrossRefGoogle Scholar
  4. [GoCoCa00]
    Gomes, H., Colominas, I., Casteleiro, E.M.: Finite element model and applications. Int. J. Numer. Meth. Eng. 00, 1–6 (2000)Google Scholar
  5. [HuRoDo02]
    Huerta, A., Roig, B., Donea, J.: Time-accurate solution of stabilized convection–diffusion–reaction equations. II: accuracy analysis and examples. Comm. Numer. Meth. Eng. 18, 575–584 (2002)MathSciNetMATHCrossRefGoogle Scholar
  6. [KuEsDa04]
    Kutluay, S., Esen A., Dag, I.: Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math. 167, 21–33 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. [OdEtAl03]
    Oden, J.T., Belytschko, T., Babuska, I., Hughes, J.R.: Research directions in computational mechanics. Comput. Meth. Appl. Mech. Eng. 192, 913–922 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. [RoSa07]
    Rodríguez–Ferran, A., Sandoval, M.L.: Numerical performance of incomplete factorizations for 3D transient convection-diffusion problems. Adv. Eng. Software 38, 439–450 (2007)Google Scholar
  9. [TaShDe07]
    Tabatabaei, A.E.A.H.H., Shakour, E., Dehghan, M.: Some implicit methods for the numerical solution of Burgers’ equation. Appl. Math. Comput. 191, 560–570 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. [TiYu11]
    Tian, Z.F., Yu, P.X.: A High-order exponential scheme for solving 1D unsteady convection–difusion equations. J. Comput. Appl. Math. 235, 2477–2491 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. [Ve04]
    Venutelli, M.: Time-stepping Padé–Petrov–Galerkin models for hydraulic jump simulation. Math. Comput. Simulat. 66, 585–604 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.State University of LondrinaLondrinaBrazil

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