Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method. (8th January 2019)

• Record Type: Journal Article
• Authors: Abazari, Reza; Yildirim, Kenan
• Other Names: Yıldırım Kenan guestEditor.; Rasa Ioan guestEditor.; Acar Tuncer guestEditor.; Abazari Reza guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 16(2019)
• Page Start: 5509
• Page End: 5521
• Publication Date: 2019-01-08
• Subjects: Crank‐Nicolson method -- solid‐liquid interface -- Sivashinsky equation -- splitting scheme -- solvability; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5454
• Languages: English
• ISSNs: 0170-4214

Abstract

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.