Notes on the numerical solution of the Benjamin equation

TL;DR

This paper introduces a new numerical scheme combining finite-element and spectral methods to solve the Benjamin equation, enabling detailed study of solitary wave dynamics and interactions.

Contribution

A novel fully discrete hybrid finite-element / spectral scheme for the Benjamin equation, validated for accuracy and stability, and used to explore wave phenomena.

Findings

Validated the numerical scheme's accuracy and stability.

Analyzed solitary wave propagation and interactions.

Explored stability of various solitary wave types.

Abstract

In this paper we consider the Benjamin equation, a partial differential equation that models one-way propagation of long internal waves of small amplitude along the interface of two fluid layers under the effects of gravity and surface tension. We solve the periodic initial-value problem for the Benjamin equation numerically by a new fully discrete hybrid finite-element / spectral scheme, which we first validate by pinning down its accuracy and stability properties. After testing the evolution properties of the scheme in a study of propagation of single - and multi-pulse solitary waves of the Benjamin equation, we use it in an exploratory mode to illuminate phenomena such as overtaking collisions of solitary waves, and the stability of single-, multi-pulse and 'depression' solitary waves.