Abstract
In this paper, a linearized three level difference scheme is derived for two-dimensional model of an alloy solidification problem called Sivashinsky equation. Further, it is proved that the scheme is uniquely solvable and convergent with convergence rate of order two in a discrete L∞-norm. At last, numerical experiments are carried out to support the theoretical claims.
Highlights
In the solidification of a dilute binary alloy, a planer solid-liquid interface is often to be instable, spontaneously assuming a cellular structure. This situation enables one to derive an asymptotic nonlinear equation which directly describes the dynamic of the onset and stabilization of cellular structure where α is a positive constant
We introduce the mathematical model for a finite difference discretization to the solution of the periodical boundary of two-dimensional Sivashinsky equation: ut + ∆2u + αu = ∆f (u), (
A semidiscrete approximation of the two dimensional Sivashinsky equation with lumped-mass method and optimal order error bounds for the piecewise linear approximation are derived in [5]
Summary
In the solidification of a dilute binary alloy, a planer solid-liquid interface is often to be instable, spontaneously assuming a cellular structure. We introduce the mathematical model for a finite difference discretization to the solution of the periodical boundary of two-dimensional Sivashinsky equation: ut + ∆2u + αu = ∆f (u),. A semi-implicit finite difference scheme and a linearized finite difference method for the Sivashinsky equation in one-dimensional have been proposed respectively in [3] [4]. A semidiscrete approximation of the two dimensional Sivashinsky equation with lumped-mass method and optimal order error bounds for the piecewise linear approximation are derived in [5]. There are many papers that have already been published to study the finite difference method for fourth-order nonlinear equation, for example [5]-[14] and so on. We investigate a linearized three level difference scheme for two-dimensional Sivashinsky equations. Some numerical examples are presented to improve the theoretical results
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