A hybridizable discontinuous Galerkin method for the Ostrovsky equation

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.