Numerical Algorithms

, Volume 71, Issue 2, pp 437–455 | Cite as

Galerkin-Chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations

Original Paper


In this paper, we present a high-order accurate method for two-dimensional semilinear parabolic equations. The method is based on a Galerkin-Chebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Optimal a priori error bound is derived in the weighted \(L^{2}_{\omega }\)-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence properties of the method.


Spectral method Block boundary value methods Semilinear parabolic equation Error estimate 

Mathematical Subject Classifications (2010)

65M12 65M60 65M70 


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  1. 1.
    Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Cambridge Philos. Soc 43, 50–67 (1947)MathSciNetMATHGoogle Scholar
  2. 2.
    Smith, G.: Numerical solution of partial differential equations: finite difference methods, 2nd edn. Oxford University Press, Oxford (1978)MATHGoogle Scholar
  3. 3.
    Van lent, J., Vandewalle, S.: Multigrid methods for implicit Runge-Kutta and boundary value method discretizations of PDEs. SIAM. J. Sci. Comput 27, 67–92 (2005)MathSciNetMATHGoogle Scholar
  4. 4.
    Mardal, K.A., Nilssen, T.K., Staff, G.A.: Order optimal preconditioners for implicit Runge-Kutta schemes applied to parabolic PDEs. SIAM. J. Sci. Comput 29, 361–375 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge Kutta methods for semilinear parabolic problems. SIAM. J. Numer. Anal 43, 1069–1090 (2005)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Hochbruck, M., Ostermann, A.: Exponential Runge-Kutta methods for parabolic problems. Appl. Numer. Math 53, 323–339 (2005)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Chawia, M.M., Al-Zanaidi, M.A., Shammeri, A.Z.: High-accuracy finite-difference schemes for the diffusion equation. Neural Parallel Sci. Comput 6, 523–535 (1998)Google Scholar
  8. 8.
    Thomas, J.W.: Numerical Partial Differential Equations:finite difference methods. Springer-Verlag, New York (1995)CrossRefMATHGoogle Scholar
  9. 9.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer-Verlag, Berlin Heidelberg (2006)MATHGoogle Scholar
  10. 10.
    Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Beach Science Publishers, Amsterdam (1998)Google Scholar
  11. 11.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)CrossRefMATHGoogle Scholar
  12. 12.
    Mohebbi, A., Dehghan, M.: High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl. Math. Model. 34, 3071–3084 (2010)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Shamsi, M., Dehghan, M.: Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. Numer. Methods Partial Differential Eq 28, 74–93 (2012)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Lakestani, M., Dehghan, M.: The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement. J. Comput. Appl. Math 235, 669–678 (2010)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Saadatmandi, A., Dehghan, M.: Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications. Int. J. Comput. Math 87, 997–1008 (2010)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Dehghan, M., Shakeri, F.: Method of lines solutions of the parabolic inverse problem with an overspecification at a point. Numer. Algorithms 50, 417–437 (2009)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Shamsi, M., Dehghan, M.: Recovering a time-dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method. Numer. Methods Partial Differential Eq 23, 196–210 (2007)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Dehghan, M., Yousefi, S.A., Rashedi, K.: Ritz-Galerkin method for solving an inverse heat conduction problem with a nonlinear source term. Inverse Probl. Sci. En 21, 500–523 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Mohebbi, A., Dehghan, M.: The use of compact boundary value method for the solution of two-dimensional Schrödinger equation. J. Comput. Appl. Math 225, 124–134 (2009)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Dehghan, M., Mohebbi, A.: High-order compact boundary value method for the solution of unsteady convection-diffusion problems. Math. Comput. Simul 79, 683–699 (2008)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Brugnano, L., Trigiante, D.: Stability properties of some BVM methods. Appl. Numer. Math 13, 201–304 (1993)Google Scholar
  22. 22.
    Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge-Kutta methods. Comput. Math. Appl 36, 269–284 (1998)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Brugnano, L.: Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems. J. Comput. Math 15, 233–252 (1997)MathSciNetMATHGoogle Scholar
  24. 24.
    Brugnano, L., Trigiante, D.: Block Boundary Value Methods for Linear Hamiltonian Systems. Appl. Math. Comput 81, 49–68 (1997)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer-Verlag, NewYork (2011)CrossRefGoogle Scholar
  26. 26.
    Shen, J.: Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials. SIAM. J. Sci. Comput 16, 74–87 (1995)MATHGoogle Scholar
  27. 27.
    Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer-Verlag, New York (2011)CrossRefGoogle Scholar
  28. 28.
    Guo, B.Y.: Spectral Methods and Their Applications. World Scietific, Singapore (1998)CrossRefMATHGoogle Scholar
  29. 29.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang T.A.: Spectral Methods in Fluid Mechanics. Springer-Verlag, New York (1988)CrossRefGoogle Scholar
  30. 30.
    Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput 21, 323–339 (1999)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Liu, W.J., Sun, J.B., Wu, B.Y.: Space-time spectral method for the two-dimensional generalized sine-Gordon equation. J. Math. Anal. Appl 427, 787–804 (2015)CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China

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