Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

• Published in: Engineering with computers Vol. 37; no. 2; pp. 1159 - 1179
• Main Authors: Dehghan, Mehdi, Shafieeabyaneh, Nasim
• Format: Journal Article
• Language: English
• Published: London Springer London 01.04.2021; Springer Nature B.V
• Subjects: Algorithms; Approximation; CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Computer Science; Computer-Aided Engineering (CAD; Control; Differential equations; Finite difference method; Finite element method; Math. Applications in Chemistry; Mathematical and Computational Engineering; Matrix methods; Meshless methods; Numerical methods; Operators (mathematics); Original Article; Parameters; Radial basis function; Robustness (mathematics); Systems Theory; Extended Fisher–Kolmogorov (EFK) equation; Nonlinear regularized long-wave (RLW) equation; Radial basis functions (RBFs); 65M70; 65M06; Shape parameter; Local meshless method; RBF–FD method; 35K55
• ISSN: 0177-0667; 1435-5663
• DOI: 10.1007/s00366-019-00877-z

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.