# 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
• 65 21

#### 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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