A hybridizable discontinuous Galerkin method for the Ostrovsky equation

TL;DR

This paper introduces a hybridizable discontinuous Galerkin method for the Ostrovsky equation, effectively handling nonlocal terms, proving stability and error estimates, and demonstrating accurate simulation of wave phenomena including solitary and peaked waves.

Contribution

The paper develops a novel HDG scheme for the Ostrovsky equation, incorporating a mixed formulation to localize nonlocal terms and providing stability and error analysis.

Findings

The scheme achieves optimal convergence rates for smooth solutions.

It accurately captures solitary wave and peakon propagation.

Numerical results confirm stability and effectiveness for various wave types.

Abstract

This paper develops the hybridizable discontinuous Galerkin (HDG) method for the Ostrovsky equation, a nonlinear dispersive wave equation featuring both third-order dispersion and a nonlocal antiderivative term with Coriolis effect. On a bounded interval, the nonlocal operator $\partial_x^{-1}$ is localized through an auxiliary variable $v$ satisfying $v_x=u$ together with an additional boundary constraint that ensures uniqueness. We employ a mixed first-order formulation to decompose the dispersive operator and to localize the nonlocal term, and we couple the resulting semi-discrete HDG scheme with a $\theta$-time stepping method for $\theta \in [1/2,1]$. We prove $L^2$-stability for suitable stabilization parameters and derive an {\it a priori} $L^2(\Omega)$ error estimate for smooth solutions that explicitly accounts for the nonlinear convective flux. Numerical examples illustrate the convergence properties and demonstrate the scheme's capability to handle smooth and non-smooth solutions, including solitary wave propagation and peaked solitary wave (peakon) propagation in the zero dispersive limit regime.