Numerical study of the 2D Sivashinsky equation for binary alloy solidification problems
https://doi.org/10.26907/2541-7746.2025.4.607-626
Abstract
Binary alloy solidification involves the transition of a liquid mixture of two metals into a solid phase and presents several complex challenges that researchers aim to address. These problems can be categorized into issues related to thermodynamics, diffusion, and macro- and microstructural evolution during the cooling process. The Sivashinsky equation is a fourth-order nonlinear partial differential equation that arises in the mathematical modeling of binary alloy solidification problems. In this article, we apply the Fourier spectral method combined with the Euler method to numerically solve the 2D Sivashinsky equation with periodic boundary conditions. A numerical study of the Sivashinsky equation is important because its analytical solution does not exist, except for trivial solutions. The error estimation of the approximate solution is provided. Furthermore, we show, both theoretically and numerically, that the proposed method preserves the decreasing mass condition of the obtained numerical solutions. Finally, to validate the theoretical results, three examples with different initial conditions are investigated.
About the Authors
R. Abazari
University of Mohaghegh Ardabili
Islamic Republic of Iran
Islamic Republic of Iran
Reza Abazari, PhD, Assistant Professor, Department of Mathematics
K. Yildirim
Mu¸s Alparslan University
Turkey
Turkey
Kenan Yildirim, PhD, Professor, Faculty of Mathematics
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For citations:
Abazari R., Yildirim K. Numerical study of the 2D Sivashinsky equation for binary alloy solidification problems. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(4):607-626. https://doi.org/10.26907/2541-7746.2025.4.607-626
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