The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

Abstract

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method.