Finite volume methods for unidirectional dispersive wave models

TL;DR

This paper develops finite volume methods for simulating unidirectional dispersive water waves, specifically using a KdV-BBM type equation, and validates the schemes through comparisons to analytical solutions and studies of wave phenomena.

Contribution

It extends finite volume methods to dispersive wave models, incorporating explicit and IMEX Runge-Kutta schemes, and analyzes invariant conservation and nonlinear wave interactions.

Findings

Validated schemes against analytical solutions

Demonstrated conservation of invariants

Simulated dispersive shock waves and solitary wave interactions

Abstract

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.