Numerical study of Sivashinsky equation using a splitting scheme based on Crank-Nicolson method

Abstract

The mathematical modeling of a planar solid-liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second-order splitting finite difference scheme, based on Crank-Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme. © 2019 John Wiley & Sons, Ltd.