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Comparison of Some Finite Difference Schemes for Boussinesq Paradigm Equation

  • Milena Dimova
  • Natalia Kolkovska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7125)

Abstract

The aim of the paper is to propose and study families of finite difference schemes for solving the Boussinesq Paradigm Equation. The nonlinear term of the equation is approximated in three different ways. We obtained a pair of implicit (with respect to the nonlinearity) families of schemes and an explicit one. All schemes have second rate of convergence in space and time. Numerical tests performed confirm our theoretical results regarding accuracy and convergence of all three schemes.

Keywords

Boussinesq Paradigm Equation finite difference method conservative schemes solitons 

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References

  1. 1.
    Bogolubsky, I.L.: Some Examples of Inelastic Soliton Interaction. Comp. Phys. Commun. 13, 149–155 (1977)CrossRefGoogle Scholar
  2. 2.
    Bratsos, A.G.: A Second Order Numerical Scheme for the Solution of the One-dimensional Boussinesq Equation. Numer. Algor. 46, 45–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chertock, A., Christov, C.I., Kurganov, A.: Central-Upwind Schemes for the Boussinesq Paradigm Equation. Computational Science and High Performance Computing IV, NNFM 113, 267–281 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Christov, C.I.: Gaussian elimination with pivoting for multidiagonal systems. Internal Report, University of Reading, 4 (1994)Google Scholar
  5. 5.
    Christov, C.I.: An Energy-consistent Dispersive Shallow-water Model. Wave Motion 34, 161–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christov, C.I., Kolkovska, N., Vasileva, D.: On the Numerical Simulation of Unsteady Solutions for the 2D Boussinesq Paradigm Equation. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 386–394. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Christov, C.I., Velarde, M.G.: Inelastic Interaction of Boussinesq Solitons. Int. J. Bifurcation & Chaos 4, 1095–1112 (1994)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kolkovska, N.T.: Convergence of Finite Difference Schemes for a Multidimensional Boussinesq Equation. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 469–476. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Kolkovska, N., Dimova, M.: A New Conservative Finite Different Scheme for Boussinesq Paradigm Equation. In: AIP (submitted, 2011)Google Scholar
  10. 10.
    Manoranjan, V.S., Mitchell, A.R., Morris, J.L.: Numerical Solutions of Good Boussinesq Equation. SIAM J. Sci. Stat. Comput. 5(4), 946–957 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Samarsky, A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)CrossRefGoogle Scholar
  12. 12.
    Samarsky, A.A., Nikolaev, E.: Numerical Methods for Grid Equations. Birkhäuser Verlag (1989) ISBN 3764322780Google Scholar
  13. 13.
    Yan, Z., Bluman, G.: New Compacton Solutions and Solitary Patterns Solutions of Nonlinearly Dispersive Boussinesq Equations. Comp. Phys. Commun. 149, 11–18 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Milena Dimova
    • 1
  • Natalia Kolkovska
    • 1
  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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