Journal of the
Korean Mathematical Society JKMS

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

MSC numbers: 35G25, 65M06, 65M12, 65M15