Skip to main content
SpringerLink
Log in

Stability analysis of a high-order finite-difference scheme for the Korteweg–de Vries equation with non-homogeneous boundaries

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we analyze the stability of a high-order finite-difference scheme for the Korteweg–de Vries (KdV) equation with non-homogeneous boundaries. We first employ a variable transformation to change the non-homogeneous boundaries to homogeneous boundaries. We then develop a fourth-order accurate finite-difference scheme for solving the transformed KdV problem. The stability, convergence and solvability of the numerical solution are analyzed. Numerical examples are given to confirm the good accuracy and the effectiveness of the present method for handling the KdV equations with non-homogeneous boundaries.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Bayarassou K, Rouatbi A, Omrani K (2020) Uniform error estimates of fourth-order conservative linearized difference scheme for a mathematical model for long wave. Int J Comput Math 97:1678–1703

    Article  MathSciNet  Google Scholar 

  • Boussinesq JV (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C R Acad Sci Paris 72:755–759

    MATH  Google Scholar 

  • Chousurin R, Mouktonglang T, Charoensawan P (2019) Fourth-order conservative algorithm for nonlinear wave propagation: the Rosenau-KdV equation. Thai J Math 17:789–803

    MathSciNet  Google Scholar 

  • Cui J, Sun Z, Wu H (2015) A high accurate and conservative difference scheme for the solution of nonlinear Schrödinger equation. Numer Math J Chin Univ 37:31–52

    MATH  Google Scholar 

  • Dehghan M (2005) On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer Method Partial Differ Equations 21(1):24–40

    Article  MathSciNet  Google Scholar 

  • Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16–30

    Article  MathSciNet  Google Scholar 

  • Dehghan M (2007) A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn 50:111–120

    Article  MathSciNet  Google Scholar 

  • Dehghan M, Abbaszadeh M (2018) Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. Comput Appl Math 37:3869–3905

    Article  MathSciNet  Google Scholar 

  • Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Method Partial Differ Equations 26(2):448–479

    Article  MathSciNet  Google Scholar 

  • Dutta R, Koley U, Risebro NH (2015) Convergence of a higher order scheme for the Korteweg-de Vries equation. SIAM J Numer Anal 53(4):1963–1983

    Article  MathSciNet  Google Scholar 

  • Dutta R, Holden H, Koley U, Risebro NH (2016) Convergence of finite difference schemes for the Benjamin-Ono equation. Numer Math 134(2):249–274

    Article  MathSciNet  Google Scholar 

  • Fermo L, Mee CVD, Seatzu S (2020) A numerical method to compute the scattering solution for the KdV equation. Appl Numer Math 149:3–16

    Article  MathSciNet  Google Scholar 

  • Gandarias ML, Bruzon MS (2012) Some conservation laws for a forced KdV equation. Nonlinear Anal Real World Appl 13(6):2692–2700

    Article  MathSciNet  Google Scholar 

  • Ghiloufi A, Kadri T (2017) Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation. Appl Anal 96:1255–1267

    Article  MathSciNet  Google Scholar 

  • Ghiloufi A, Omrani K (2018) New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer Method Partial Differ Equations 34:451–500

    Article  MathSciNet  Google Scholar 

  • Gorsky J, Himonas AA, Holliman C, Petronilho G (2013) The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces. J Math Anal Appl 405(2):349–361

    Article  MathSciNet  Google Scholar 

  • Guha P (2003) Moving space curve equations and a family of coupled KdV type systems. Chaos Soliton Fract 15(1):41–46

    Article  MathSciNet  Google Scholar 

  • Holden H, Koley U, Risebro NH (2015) Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation. IMA J Numer Anal 35(3):1047–1077

    Article  MathSciNet  Google Scholar 

  • Koley U (2012) Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation. Cent Eur J Math 10(1):173–187

    Article  MathSciNet  Google Scholar 

  • Koley U (2016) Finite difference schemes for the Korteweg-de Vries-Kawahara equation. Int J Numer Anal Mod 13(3):344–367

    MathSciNet  MATH  Google Scholar 

  • Kong D, Xu Y, Zheng Z (2019) A hybrid numerical method for the KdV equation by finite difference and sinc collocation method. Appl Math Comput 355:61–72

    MathSciNet  MATH  Google Scholar 

  • Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal. Philos Mag 39(240):422–443

    Article  MathSciNet  Google Scholar 

  • Liu GR, Gu YT (2004) Boundary meshfree methods based on the boundary point interpolation methods. Eng Anal Bound Elements 28(5):475–487

    Article  Google Scholar 

  • Liu HL, Yan J (2006) A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect. J Comput Phys 215(1):197–218

    Article  MathSciNet  Google Scholar 

  • Morton KW, Mayers DF (1994) Numerical solution of partial differential equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Shen J (2003) A new dual-Petrov–Galerkin method for third and higher odd-order differential equations: application to the KdV equation. SIAM J Numer Anal 41(5):1595–1619

    Article  MathSciNet  Google Scholar 

  • Tamang N, Wongsaijai B, Mouktonglang T, Poochinapan K (2020) Novel algorithm based on modification of Galerkin finite element method to general Rosenau-RLW equation in (2+1)-dimensions. Appl Numer Math 148:109–130

    Article  MathSciNet  Google Scholar 

  • Wang XF, Dai W (2018) A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation. J Comput Appl Math 330:295–306

    Article  MathSciNet  Google Scholar 

  • Wang XF, Dai W, Usman M (2021) A high-order accurate finite difference scheme for the KdV equation with time-periodic boundary forcing. Appl Numer Math 160:102–121

    Article  MathSciNet  Google Scholar 

  • Wongsaijai B, Poochinapan K (2014) A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and the Rosenau-CRLW equation. Appl Math Comput 245:289–304

    MathSciNet  MATH  Google Scholar 

  • Zhang X, Zhang P (2018) A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation. Appl Math Comput 339:535–545

    Article  MathSciNet  Google Scholar 

  • Zhou Y (1990) Application of discrete functional analysis to the finite difference methods. International Academic Publishers, Beijing

    Google Scholar 

Download references

Acknowledgements

The authors were supported in part supported by the Natural Science Foundation of Fujian Province, China (No: 2020J01796). The authors also thank the reviewers and editors for their very helpful comments and suggestions which greatly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, Fujian, People’s Republic of China

    Hong Cheng & Xiaofeng Wang

Authors
  1. Hong Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaofeng Wang.

Additional information

Communicated by Corina Giurgea.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, H., Wang, X. Stability analysis of a high-order finite-difference scheme for the Korteweg–de Vries equation with non-homogeneous boundaries. Comp. Appl. Math. 40, 49 (2021). https://doi.org/10.1007/s40314-021-01443-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01443-4

Keywords

Mathematics Subject Classification

Search

Navigation