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 [6] to confirm the accuracy and efficiency of the presented method.

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

“…The Equations (14)-(16) preserve the cone condition for each advanced time. Furthermore, the scheme (14) unconditionally preserves the fixed point and the geometric property of the true flows of the original equation (see [25], Theorem 1,2).…”

“…The goal is to take advantage of both meshless method and GNI method for numerical approximations obtained by this combination of the Lie-Group method based on radial basis functions (LG-RBFs). This method has been recently applied to the Heat equation [24] and high-dimensional generalized Benjamin-Bona-Mahony-Burgers' equation [25] .…”

Numerical integrators based on the Magnus expansion for nonlinear dynamical systems

“…We see then that, by applying Magnus expansion on augmented systems results in higher order GPS. These schemes could be applied, e.g., to get higher order approximations in combination with radial basis functions (RBFs) for PDEs [26,27].…”

Meshless formulation to two‐dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

“…Table 3 shows the error and order of convergence by increasing t with fixed N = 256 at t = 1. The maximum absolute error of this computed LG-RBF [35] 8.7045 × 10 −6 5.7507 × 10 −6…”

“…In special case, if = 0, = 1 and = −1 then the problem acquired to the regularized long-wave equation (RLWE) that was studied as an improvement of the Korteweg-de Vries (KdV) equation for modeling long surface gravity waves. Various methods (analytical and numerical) have been applied for solving the BBMB equation [32][33][34][35][36][37][38][39]. Our goal is to deal with the GBBMB through the SBM in order to acquire more accurate results in comparison with other methods.…”