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

## TL;DR

This paper develops a Petrov-Galerkin finite element method using cubic shape and quadratic B-spline weight functions to accurately and stably approximate solutions of the nonlinear generalized regularized long wave equation, validated through multiple test problems.

## Contribution

It introduces a novel Petrov-Galerkin scheme with specific spline functions for the GRLW equation, providing stability analysis and demonstrating high accuracy in numerical simulations.

## Key findings

- The scheme is unconditionally stable based on Fourier analysis.
- Numerical results show high accuracy compared to existing methods.
- The method effectively captures solitary wave interactions and evolution.

## Abstract

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear generalized regularized long wave (GRLW) equation by Petrov--Galerkin method in which the element shape functions are cubic and weight functions are quadratic B-splines. The suggested method is performed to three test problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate accuracy of such a spatial approximation. Then Fourier stability analysis of the linearized scheme shows that it is unconditionally stable. To verify practicality and robustness of the new scheme error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated. The obtained numerical results are compared with other published results and shown to be precise and effective.