Numerical study of the 2d sivashinsky equation for binary alloy solidification problems

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. © 2025, Kazan Federal University. All rights reserved.