A hybridizable discontinuous Galerkin method for the Ostrovsky equation

TL;DR

The paper presents a high-order HDG method for the Ostrovsky equation, tackling the third-order dispersion and a nonlocal inverse-derivative term by a mixed first-order formulation with an auxiliary variable and a boundary gauge to ensure uniqueness. A θ-time discretization ($\theta\in[1/2,1]$) is analyzed for stability and accuracy, and an $L^2$-error bound $\|u-u_h\|_{L^2(\Omega)} \le C h^{k+1/2}$ is established for smooth solutions. Numerical experiments confirm optimal spatial convergence, robust solitary-wave and peakon handling, and correct asymptotic behavior in the $\beta\to0$ limit toward OH dynamics, with efficient static condensation reducing global degrees of freedom. The work lays groundwork for extensions to KP-type models in higher dimensions and provides a framework combining locality, conservation, and high-order accuracy for dispersive nonlocal PDEs.

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 $θ$-time stepping method for $θ\in [1/2,1]$. We prove $L^2$-stability for suitable stabilization parameters and derive an {\it a priori} $L^2(Ω)$ 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.

Figures (2)

• Figure 1: Solitary-wave propagation with $N_e=256$, $k=2$, $\Delta t=0.05$, $T=20$. Shown are $u_h(\cdot,t)$ at $t=0,5,10,15,20$ and the translated reference $u_{\rm ref}(\cdot,T)=U(\cdot-c_wT)$.
• Figure 2: Peakon solution and the limit $\beta\to0$. Shown are the initial condition $u_0$ from \ref{['eq:peakon_u0']}, the OH reference profile $u_{\mathrm{OH}}(\cdot,T)$ given by \ref{['eq:peakon_exact']}, and periodic HDG solutions $u_h(\cdot,T)$ for several values of $\beta$.

Theorems & Definitions (12)

• Theorem 3.2: Stability
• proof
• Remark 3.3: Conservative property for periodic problems with $\beta>0$ and $\gamma>0$
• Remark 3.4: Case $\beta < 0$
• Lemma 4.1: Inverse inequalities Ciarlet1978FemXuShu2007
• Lemma 4.2: Interpolation inequality Ciarlet1978FemXuShu2007
• Theorem 4.3
• proof
• Theorem 5.1: Fully discrete stability
• proof