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A fourth-order accurate finite difference scheme for the extended-Fisher-Kolmogorov equation

Khaled Omrani|Tlili Kadri
대한수학회 (The Korean Mathematical Society)|2018.01|대한수학회보 (Bulletin of the KMS)|제55권 제1호|pp. 297-310

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참고문헌

키워드

uniquenessexistenceextended Fisher-Kolmogorov equationdifference schemehigh-order convergence
주제분류수학

초록

In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}$-norm. Some numerical examples are given in order to validate the theoretical results.

Abstract

In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}$-norm. Some numerical examples are given in order to validate the theoretical results.

참고문헌 (27)

  1. G. Ahlers, "Vortex-front propagation in rotating Couette-Taylor Tow", Phys. Rev. Lett., 50, pp.1583~1586 (1983)[DOI]
  2. G. D. Akrivis, "Finite difference discretization of Kuramoto-Sivashinsky", Numer. Math., 63(1), pp.1~11 (1992)[DOI]
  3. D. G. Aronson, "Multidimensional nonlinear diffusion arising in population genetics", Adv. in Math., 30(1), pp.33~76 (1978)[DOI]
  4. F. E. Browder, "Existence and uniqueness theorems for solutions of nonlinear boundary value problems", Proc. Sympos. Appl. Math., XVII, pp.24~49 (1965), Amer. Math. Soc.
  5. P. Coullet, "Nature of spatial chaos", Phys. Rev. Lett., 58(5), pp.431~434 (1987)[DOI]
  6. G. T. Dee, "Bistable systems with propagating fronts leading to pattern formation", Phys. Rev. Lett., 60, pp.2641~2644 (1988)[DOI]
  7. M. Dehghan, "Fourth-order compact solution of the nonlinear Klein-Gordon equation", Numer. Algorithms, 52(4), pp.523~540 (2009)[DOI]
  8. M. Dehghan, "A compact split-step finite difference method for solving the nonlinear Schrodinger equations with constant and variable coecients", Comput. Phys. Commu., 181(1), pp.43~51 (2010)[DOI]
  9. D. He, "Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau-Kawahara-RLW equation with generalized Novikov type perturbation", Nonlinear Dynam., 85(1), pp.479~498 (2016)[DOI]
  10. D. He, "A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation", Appl. Math. Comput., 271, pp.323~336 (2015)
  11. R. M. Hornreich, "Critical behaviour at the onset of k-space instability at the <TEX>$\lambda$<", TEX> line, 1975, pp.35, 1678~1681
  12. N. Khiari, "Finite difference approximate solutions for the Cahn-Hilliard equation", Numerical Methods for Partial Differen-tial Eqs., 23(2), pp.437~455 (2007)[DOI]
  13. N. Khiari, "Finite difference discretization of the extended Fisher Kolmogorov equation in two dimensions", Comput. Math. Appl., 62, pp.4151~4160 (2011)[DOI]
  14. A. Mohebbi, "High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods", Math. Comput. Mod-elling, 51(5-6), pp.537~549 (2010)[DOI]
  15. A. Mohebbi, "High-order compact solution of the one-dimensional heat and advectiondi ffusion equations", Appl. Math. Model., 34(10), pp.3071~3084 (2010)[DOI]
  16. K. Omrani, "A second order splitting method for a finite difference scheme for the Sivashinsky equation", Appl. Math. Lett., 16(3), pp.441~445 (2003)[DOI]
  17. K. Omrani, "Numerical methods and error analysis for the nonlinear Sivashinsky equation", Appl. Math. Comput., 189(1), pp.949~962 (2007)
  18. K. Omrani, "A new conservative finite difference scheme for the Rosenau equation", Appl. Math. Comput., 201(1-2), pp.35~43 (2008)
  19. X. Pan, "On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation", Appl. Math. Model., 36(8), pp.3371~3378 (2012)[DOI]
  20. X. Pan, "Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme", Math. Probl. Eng., 2012 (2012)
  21. L. J. Tracius Doss, "An <TEX>$H^1$<", TEX>-Galerkin Mixed Finite Element Method For The Extended Fisher-Kolmogorov Equation, 2012, pp.3(4), 460~485
  22. T. Wang, "Convergence of an eighth-order compact difference scheme for the nonlinear Schrodinger equation", Adv. Numer. Anal., 2012, pp.24~ (2012)
  23. T. Wang, "Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schrodinger equation", J. Math. Anal. Appl., 412(1), pp.155~167 (2014)[DOI]
  24. T. Wang, "Fourth-order compact and energy conservative difference schemes for the nonlinear Schrodinger equation in two dimensions", J. Comput. Phys., 243, pp.382~399 (2013)[DOI]
  25. T. Wang, "Point-wise errors of two conservative difference schemes for the Klein-Gordon-Schrodinger equation", Commun. Nonlinear Sci. Numer. Simul., 17(12), pp.4565~4575 (2012)[DOI]
  26. Y. Zhou, "Application of Discrete Functional Analysis to the Finite Difference Methods" (1990), International Academic Publishers
  27. G. Zhu, "Experiments on director waves in nematic liquid crystals", Phys. Rev. Lett., 49, pp.1332~1335 (1982)[DOI]

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