Modeling and Numerical Simulation of Material Science, 2016, 6, 1-9
Published Online January 2016 in SciRes. http://www.scirp.org/journal/mnsms
http://dx.doi.org/10.4236/mnsms.2016.61001
How to cite this paper: Rouis, M. and Omrani, K. (2016) On The Numerical Solution of Two Dimensional Model of an Alloy
Solidification Problem. Modeling and Numerical Simulation of Material Science, 6, 1-9.
http://dx.doi.org/10.4236/mnsms.2016.61001
On The Numerical Solution of Two
Dimensional Model of an Alloy
Solidification Problem
Moeiz Rouis, Khaled Omrani
Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Sousse, Tunisia
Received 30 December 2015; accepted 25 January 2016; published 28 January 2016
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
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.
Keywords
Solidification Problem, Sivashinsky Equation, Linearized Difference Scheme, Solvability,
Convergence
1. Introduction
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 di-
rectly describes the dynamic of the onset and stabilization of cellular structure
( )
4
4
2 0,
u u u
u u
t x x x
α
∂ ∂ ∂ ∂
+ + − + =
∂ ∂ ∂ ∂
(1.1)
where
α is a positive constant, (see [1] [2]). Equation (1.1) is referred as the Sivashinsky equation.
In this article, we introduce the mathematical model for a finite difference discretization to the solution of the
periodical boundary of two-dimensional Sivashinsky equation:
( ) ( )
2 2
, , , 0 ,
t
u u u f u xy t T α +∆ + =∆ ∈ < ≤ (1.2)