Fourier Spectral Methods for Numerical Solving of the Kuramoto-Sivashinsky Equation

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Gentian Zavalani
Gentian Zavalani
1 Polytechnic University of Tirana/ Albania

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In this paper I presented a numerical technique for solving Kuramoto-Sivashinsky equation, based on spectral Fourier methods. This equation describes reaction diffusion problems, and the dynamics of viscous-fuid films flowing along walls. After we wrote the equation in Furie space, we get a system. In this case, the exponential time differencing methods integrate the system much more accurately than other methods since the exponential time differencing methods assume in their derivation that the solution varies slowly in time. When evaluating the coefficients of the exponential time differencing and the exponential time differencing Runge Kutta methods via the”Cauchy integral”. All computational work is done with Matlab package.

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References

  1. Gregory Beylkin,James Keiser,Lev Vozovoi (1998). A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs.
  2. J Certaine The Solution of Ordinary Dierential Equations with Large Time Constants.
  3. Friedli (1978). Generalized Runge-Kutta Methods for the Solution of Stiff Dierential Equations.
  4. S Norsett (1969). An A-Stable Modication of the Adams-Bashforth Methods.
  5. C Klein (2008). Odd-Order One-Dimensional Equations: Korteweg-de Vries, Compacton, Nonlinear Dispersion, and Harry Dym Models.
  6. A Kassam,L Trefethen (2005). Fourth-Order Time Stepping for Stiff PDEs.
  7. R Burden,J Faires (2001). Numerical Analysis.
  8. M Hochbruck,A Ostermann (2005). Explicit Exponential Runge-Kutta Methods for Semi-linear Parabolic Problems.
  9. S Cox,P Matthews (2002). Exponential Time Differencing for Stiff Systems.
  10. Arnont Saengsiritongchai (2004). Finite integration method using chebyshev polynomials for solving time-dependent linear partial differential equations and linear fractional order differential equations.

Funding

No external funding was declared for this work.

Conflict of Interest

The authors declare no conflict of interest.

Ethical Approval

No ethics committee approval was required for this article type.

Data Availability

Not applicable for this article.

Gentian Zavalani. 2014. \u201cFourier Spectral Methods for Numerical Solving of the Kuramoto-Sivashinsky Equation\u201d. Global Journal of Research in Engineering - I: Numerical Methods GJRE-I Volume 14 (GJRE Volume 14 Issue I1): .

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Journal Specifications

Crossref Journal DOI 10.17406/gjre

Print ISSN 0975-5861

e-ISSN 2249-4596

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v1.2

Issue date

June 30, 2014

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English

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In this paper I presented a numerical technique for solving Kuramoto-Sivashinsky equation, based on spectral Fourier methods. This equation describes reaction diffusion problems, and the dynamics of viscous-fuid films flowing along walls. After we wrote the equation in Furie space, we get a system. In this case, the exponential time differencing methods integrate the system much more accurately than other methods since the exponential time differencing methods assume in their derivation that the solution varies slowly in time. When evaluating the coefficients of the exponential time differencing and the exponential time differencing Runge Kutta methods via the”Cauchy integral”. All computational work is done with Matlab package.

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Fourier Spectral Methods for Numerical Solving of the Kuramoto-Sivashinsky Equation

Gentian Zavalani
Gentian Zavalani Polytechnic University of Tirana/ Albania

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