A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline

Abstract

In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms and and the conservative quantities , and are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

“…Wang et al offered a mesh‐free method for the GRLW equation based on the moving least‐squares approximation. Karakoç and Zeybek have obtained solitary‐wave solutions of the GRLW equation by using septic B‐spline collocation and cubic B‐spline lumped Galerkin method. Numerical solutions of the GRLW equation have been obtained by Soliman using He's variational iteration method.…”

“…They are spectacular mathematical instrument for numerical approximations because of their numerous popular specialties. One kind of splines, noted as B‐splines, has been used in obtaining the numerical solution of the GRLW equation . Assemblies of B‐splines are used as trial functions in the Petrov–Galerkin methods.…”

A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation

“…Mohammadi [32] obtained a numerical solution to the nonlinear GRLW equation using collocation algorithm based on exponential B-spline nite elements. Zeybek and Karako c used a nite element method with B-splines to solve the GRLW equation [33,34]. Lately, collocation scheme based on B-spline nite elements was investigated for solving the Complex Modi ed Korteweg-de Vries (CMKdV), the generalized nonlinear Schrodinger (GNLS) equation, and generalized Burgers-Fisher and Burgers-Huxley equations [35][36][37].…”

Computational Analysis of Shallow Water Waves with Korteweg-de Vries Equation

“…Some of these algorithms are exp-function method, G 0 =G-expansion scheme [14,15], method of undetermined coe cients, semi-inverse variational principle, Galerkin method [16,17], Petrov-Galerkin method [18], collocation method [19][20][21], and several others [22][23][24]. Also, the solutions to some fractional di erential equations have been discussed [25,26].…”