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Comparison of Two Numerical Approaches to Boussinesq Paradigm Equation

  • Milena Dimova
  • Daniela Vasileva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

In order to study the time behavior and structural stability of the solutions of Boussinesq Paradigm Equation, two different numerical approaches are designed. The first one (A1) is based on splitting the fourth order equation to a system of a hyperbolic and an elliptic equation. The corresponding implicit difference scheme is solved with an iterative solver. The second approach (A2) consists in devising of a finite difference factorization scheme. This scheme is split into a sequence of three simpler ones that lead to five-diagonal systems of linear algebraic equations. The schemes, corresponding to both approaches A1 and A2, have second order truncation error in space and time. The results obtained by both approaches are in good agreement with each other.

Keywords

Numerical Approach Coarse Grid Boussinesq Equation Quadratic Nonlinearity Factorize Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Christov, C.I.: An Energy-consistent Dispersive Shallow-water Model. Wave Motion 34, 161–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boussinesq, J.V.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées 17, 55–108 (1872)zbMATHGoogle Scholar
  3. 3.
    Christov, C.I., Choudhury, J.: Perturbation Solution for the 2D Boussinesq Equation. Mech. Res. Commun. 38, 274–281 (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Christov, C.I., Todorov, M.T., Christou, M.A.: Perturbation Solution for the 2D Shallow-water Waves. In: AIP Conference Proceedings, vol. 1404, pp. 49–56 (2011)Google Scholar
  5. 5.
    Christov, C.I.: Numerical Implementation of the Asymptotic Boundary Conditions for Steadily Propagating 2D Solitons of Boussinesq Type Equations. Math. Comp. Simulat. 82, 1079–1092 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christou, M.A., Christov, C.I.: Fourier-Galerkin Method for 2D Solitons of Boussinesq Equation. Math. Comput. Simul. 74, 82–92 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    Christov, C.I., Kolkovska, N., Vasileva, D.: Numerical Investigation of Unsteady Solutions for the 2D Boussinesq Paradigm Equation. 5th Annual Meeting of the Bulgarian Section of SIAM. In: BGSIAM 2010 Proceedings, pp. 11–16 (2011)Google Scholar
  10. 10.
    Ames, W.F.: Nonlinear Partial Differential Equations in Engineering. Academic Press (1965)Google Scholar
  11. 11.
    Christov, C.I., Velarde, M.G.: Inelastic interaction of Boussinesq solitons. J. Bifurcation & Chaos 4, 1095–1112 (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kolkovska, N.: 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
  13. 13.
    Dimova, M., Kolkovska, N.: Comparison of Some Finite Difference Schemes for Boussinesq Paradigm Equation. In: Adam, G., Buša, J., Hnatič, M. (eds.) MMCP 2011. LNCS, vol. 7125, pp. 215–220. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Kolkovska, N., Dimova, M.: A New Conservative Finite Difference Scheme for Boussinesq Paradigm Equation. Cent. Eur. J. Math. 10(3), 1159–1171 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kolkovska, N.: Two Families of Finite Difference Schemes for Multidimensional Boussinesq Equation. In: AIP Conference Series, vol. 1301, pp. 395–403 (2010)Google Scholar
  16. 16.
    van der Vorst, H.: Iterative Krylov Methods for Large Linear Systems. Cambridge Monographs on Appl. and Comp. Math. 13 (2009)Google Scholar
  17. 17.
    Samarskii, A.: The Theory of Difference Schemes. Marcel Dekker Inc. (2001)Google Scholar
  18. 18.
    Samarskii, A.A., Nikolaev, E.: Numerical Methods for Grid Equations. Birkhäuser Verlag (1989)Google Scholar
  19. 19.
    Christov, C.I.: Gaussian Elimination with Pivoting for Multidiagonal Systems. Internal Report, University of Reading 4 (1994)Google Scholar
  20. 20.
    Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruyter (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Milena Dimova
    • 1
  • Daniela Vasileva
    • 1
  1. 1.Institute of Mathematics and InformaticsBulgarian Acad. Sci.SofiaBulgaria

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