DOI QR코드

DOI QR Code

ERROR ESTIMATES OF NONSTANDARD FINITE DIFFERENCE SCHEMES FOR GENERALIZED CAHN-HILLIARD AND KURAMOTO-SIVASHINSKY EQUATIONS

  • Choo, Sang-Mok (School of Electrical Engineering University) ;
  • Chung, Sang-Kwon (Department of Mathematics Education Seoul National University) ;
  • Lee, Yoon-Ju (Department of Mathematics Education Seoul National University)
  • Published : 2005.11.01

Abstract

Nonstandard finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with periodic boundary conditions, which are of the type $$U_t\;+\;\frac{{\partial}^2}{{\partial}x^2} g(u,\;U_x,\;U_{xx})\;=\;\frac{{\partial}^{\alpha}}{{\partial}x^{\alpha}}f(u,\;u_x),\;{\alpha}\;=\;0,\;1,\;2$$. Stability and error estimate of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem. Three examples are provided to apply the nonstandard finite difference schemes.

Keywords

nonstandard finite difference scheme;Cahn-Hilliard equation;Kuramoto-Sivashinsky equation;Neumann boundary condition;periodic boundary condition;Lax-Richtmyer equivalence theorem

References

  1. R. P. Agarwal, Difference equations and inequalities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 155, Theory, Methods, and Applications, Marcel Dekker Inc., New York, 1992
  2. G. D. Akrivis, Finite element discretization of the Kuramoto-Sivashinsky equation, Numerical Analysis and Mathematical Modelling, Banach Center Publ., vol. 29, Polish Acad. Sci., Warsaw, 1994, pp. 155-163
  3. S. M. Choo and S. K. Chung, Conservative nonlinear difference scheme for the Cahn-Hilliard equation, Comput. Math. Appl. 36 (1998), no. 7, 31-39
  4. S. M. Choo, S. K. Chung, and K. I. Kim, Conservative nonlinear difference scheme for the Cahn-Hilliard equation. II, Comput. Math. Appl. 39 (2000), no. 1- 2, 229-243 https://doi.org/10.1016/S0898-1221(99)00326-0
  5. S. M. Choo, S. K. Chung, and Y. J. Lee, A Conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient, Appl. Numer. Math. 51 (2004), no. 2-3, 207-219 https://doi.org/10.1016/j.apnum.2004.02.006
  6. C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), no. 2, 97-128 https://doi.org/10.1093/imamat/38.2.97
  7. C. M. Elliott and D. A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal. 26 (1989), no. 4, 884-903 https://doi.org/10.1137/0726049
  8. C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal. 96 (1986), no. 4, 339-357
  9. D. Furihata, A stable and conservative finite difference scheme for the Cahn- Hilliard equation, Numer. Math. 87 (2001), no. 4, 675-699 https://doi.org/10.1007/PL00005429
  10. A. V. Manickam, K. M. Moudgalya, and A. K. Pani, Second-order splitting combined with orthogonal cubic spline collocation method for the Kuramoto- Sivashinsky equation, Comput. Math. Appl. 35 (1998), no. 6, 5-25 https://doi.org/10.1016/S0898-1221(98)00013-3
  11. J. C. Lopez Marcos and J. M. Sanz-Serna, Stability and convergence in numerical analysis. III. Linear investigation of nonlinear stability, IMA J. Numer. Anal. 8 (1988), no. 1, 71-84 https://doi.org/10.1093/imanum/8.1.71
  12. R. E. Mickens, Applications of nonstandard finite difference schemes, World Scientific, New Jersey, 2000
  13. T. Ortega and J. M. Sanz-Serna, Nonlinear stability and convergence of finite- difference methods for the 'good' Boussinesq equation, Numer. Math. 58 (1990), no. 2, 215-229 https://doi.org/10.1007/BF01385620
  14. E. Tadmor, The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 17 (1986), no. 4, 884-893 https://doi.org/10.1137/0517063
  15. D. Furihata,, Finite difference schemes for $\frac{{\partial}u}{{\partial}t}=(\frac{{\partial}}{{\partial}x})^{\alpha}\frac{{\partial}G}{{\partial}u}$. J. Comput. Phys. 181-205
  16. G. D. Akrivis, Finite difference discretization of the Kuramoto-Sivashinsky equation, Numer. Math. 63 (1992), no. 1, 1-11 https://doi.org/10.1007/BF01385844

Cited by

  1. Weak solutions for a class of metaparabolic equations vol.87, pp.8, 2008, https://doi.org/10.1080/00036810802369223
  2. Nonstandard finite difference methods: recent trends and further developments vol.22, pp.6, 2016, https://doi.org/10.1080/10236198.2016.1144748