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

摘要

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.