Finite element method for extended KdV equations

TL;DR

This paper applies the finite element method to solve extended KdV equations modeling shallow water waves, demonstrating its effectiveness in capturing soliton dynamics and presenting new results for various bottom shapes.

Contribution

It introduces a finite element approach with Petrov-Galerkin formulation for extended KdV equations, including new results for different bottom geometries.

Findings

FEM accurately models soliton wave dynamics

New results for wave behavior over varied bottom shapes

Method suitable for incorporating stochastic effects

Abstract

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.