Approximate solutions to the Allen-Cahn equation using rational radial basis functions method

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 04 شهریور 1401   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.76010.1123

نویسندگان

Mansour Shiralizadeh^{* 1}؛ Amjad Alipanah¹؛ Maryam Mohammadi²

¹Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran.

²Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.

چکیده

We apply the rational radial basis functions (RRBFs) method to solve the Allen--Cahn (A.C) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by the RRBFs method. Then we apply an explicit, fourth-order Runge--Kutta method to advance the resulting semi-discrete system in time.
It is well known that the A.C equation has a nonlinear stability feature, meaning that the free-energy functional is reduced by time. The presented method maintains the total energy reduction property of the A.C equation. In the end,  five examples to confirm the efficiency and accuracy of the proposed method are provided.

کلیدواژه‌ها

Allen-Cahn equation؛ RBFs؛ Rational RBFs method؛ Runge-Kutta method