Skip to main content
SpringerLink
Log in

Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h 4 + k 4 + τ 2 + τ 2 h 2 + τ 2 k 2) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Strauss, W.: Nonlinear wave equations. CBMS 73, AMS (1989)

  2. Mohanty, R.K.: An unconditionally stable difference scheme for the one space dimensional linear hyperbolic equation. Appl. Math. Lett. 17, 101–105 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gao, F., Chi, C.: Unconditionally stable difference schemes for a one-space dimensional linear hyperbolic equation. Appl. Math. Comput. 187, 1272–1276 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Mohebbi, A., Dehghan, M.: High order compact solution of the one-space dimensional linear hyperbolic equation. Numer. Methods. Partial. Differ. Equ. 24, 1222–1235 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Raggett, G.F., Wilson, P.D.: A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique. J. Inst. Math. Appl. 14, 75–77 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rashidinia, J., Jalilian, R., Kazemi, V.: Spline methods for the solution of hyperbolic equations. Appl. Math. Comput. 190, 882–886 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Ding, H., Zhang, Y.: A new unconditionally stable compact difference scheme of for the 1D linear hyperbolic equation. Appl. Math. Comput. 207, 236–241 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Liu, H.W., Liu, L.B.: An unconditionally stable spline difference scheme of O(k 2 + h 4) for solving the second order 1D linear hyperbolic equation. Math. Comput. Model. 49, 1985–1993 (2009)

    Article  MATH  Google Scholar 

  9. Mohanty, R.K., Gopal, V.: High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations. Appl. Math. Comput. 218(8), 4234–4244 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Mohanty, R.K., Gopal, V.: An off-step discretization for the solution of 1D mildly nonlinear wave equations with variable coefficients. J. Adv. Res. Sci. Comput. 04(02), 1–13 (2012)

    Google Scholar 

  11. Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Comput. Phys. Comm. 181(1), 78–91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rashidinia, J., Mohammadi, R.: Tension spline solution of nonlinear sine-Gordon equation. Numer. Algor. 56(1), 129–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dehghan, M., Mohebbi, A.: High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. Numer. Methods Partial Differ. Equ. 25, 232–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mittal, R.C., Bhatia, R.: Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput. 220, 496–506 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Dosti, M., Nazemi, A.: Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation. J. Inf. Comput. Sci. 7(2), 083–090 (2012)

    Google Scholar 

  16. Mohanty, R.K., Jain, M.K.: An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation. Numer. Methods. Partial Differ. Equ. 17(6), 684–688 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mohanty, R.K.: An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions. Appl. Math. Comput. 152, 799–806 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Ding, H.F., Zhang, Y.X.: A new fourth order compact finite difference scheme for the two-dimensional second order hyperbolic equations. J. Comput. Appl. Math 230, 626–632 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dehghan, M., Mohebbi, A.: High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. Numer. Methods. Partial Differ. Equ. 25, 232–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dehghan, M., Shokri, A.: A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions. Numer. Methods Partial Differ. Equ. 25, 494–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dehghan, M., Salehi, R.: A method based on meshless approach for the numerical solution of the two space dimensional hyperbolic telegraph equation. Math. Method. Appl. Sci. 35(10), 1220–1233 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Piperno, S.: Symplectic local time-stepping in non-dissipative dgtd methods applied to wave propagation problems. ESAIM: Math. Modelling. Numer. Analysis 5, 815–841 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shi, D.Y., Li, Z.Y.: Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. Applied Mathematics - A Journal of the Chinese Universities 4, 455–462 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chabassier, J., Joly, P.: Energy preserving schemes for nonlinear hamiltonian systems of wave equations: application to the vibrating piano string. Computer Methods. Appl. Mechanic. Engineering 45, 2779–2795 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chawla, M.M., Al-Zanaidi, M.A.: A linearly implicit one-step time integration scheme for nonlinear hyperbolic equations. Intern. J. Computer Math. 76, 349–361 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chawla, M.M., Al-Zanaidi, M.A.: A linearly implicit one-step time integration scheme for nonlinear hyperbolic equations in two space dimensions. Intern. J. Computer Math. 80(3), 357–365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Djidjeli, K., Price, W.G., Twizell, E.H.: Numerical solutions of a damped sine-Gordon equation in two space variables. J. Eng. Math. 29, 347–369 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dehghan, M., Shokri, A.: A numerical method for solution of the two dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simulation 79, 700–715 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jiwari, R., Pandit, S., Mittal, R.C.: Numerical simulation of two dimensional sine-Gordon solitons by differential quadrature method. Comput. Phys. Commun. 183, 600–616 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central Tehran Branch, Islamic Azad University, 13185.768, Tehran, Iran

    Homa Zadvan & Jalil Rashidinia

Authors
  1. Homa Zadvan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jalil Rashidinia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Rashidinia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zadvan, H., Rashidinia, J. Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term. Numer Algor 74, 289–306 (2017). https://doi.org/10.1007/s11075-016-0149-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0149-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation