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http://dx.doi.org/10.4134/JKMS.2005.42.6.1121

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)
Publication Information
Journal of the Korean Mathematical Society / v.42, no.6, 2005 , pp. 1121-1136 More about this Journal
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;
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