A hybridizable discontinuous Galerkin method for the Ostrovsky equation

1. Problem Statement

The paper addresses the numerical approximation of the Ostrovsky equation, a nonlinear dispersive wave equation modeling weakly nonlinear long surface waves in a rotating frame of reference. The equation is given by:
$$\partial_t u - \beta \partial_x^3 u + f(u)_x - \gamma \partial_x^{-1} u = 0$$
where:

• $u(x,t)$ is the wave amplitude.
• $f(u) = \frac{\alpha}{2}u^2$ represents nonlinear convection.
• $\beta \partial_x^3 u$ is the third-order dispersion term (KdV-type).
• $\gamma \partial_x^{-1} u$ is a nonlocal antiderivative term representing the Coriolis effect due to background rotation ($\gamma > 0$).

Key Challenges:

1. Nonlocality: The term $\partial_x^{-1}$ (defined as $\int_{+\infty}^x g(s) ds$ in the whole line, or requiring a boundary constraint on a bounded domain) introduces significant computational complexity.
2. High-Order Derivatives: The third-order spatial derivative requires high-order continuity or mixed formulations in standard finite element methods.
3. Nonlinearity: The convective flux $f(u)_x$ complicates stability and error analysis.
4. Singular Limits: The equation reduces to the Ostrovsky-Hunter (OH) equation when $\beta \to 0$, which supports non-smooth "peakon" solutions (solutions with discontinuous slopes), posing difficulties for high-order methods.

2. Methodology

The authors propose a Hybridizable Discontinuous Galerkin (HDG) method, which combines the flexibility of Discontinuous Galerkin (DG) methods with the efficiency of hybridization.

A. Mixed First-Order Formulation

To handle the third-order derivative and the nonlocal term, the authors introduce auxiliary variables to decompose the equation into a first-order system:

1. Dispersive terms: Let $q = u_x$ and $p = \beta q_x$.
2. Nonlocal term: Introduce $v$ such that $v_x = u$.
3. Localization: On a bounded interval $\Omega = (x_L, x_R)$, the operator $\partial_x^{-1}$ is not unique without a constraint. The authors impose the boundary condition $v(x_R, t) = 0$ to ensure uniqueness.

The resulting system is:
$$\begin{aligned} q &= u_x \\ p &= \beta q_x \\ v_x &= u \\ u_t - p_x + f(u)_x - \gamma v &= 0 \end{aligned}$$

B. HDG Discretization

• Spatial Discretization: The domain is partitioned into elements. Polynomial spaces of degree $k$ are used for the interior variables $(u_h, q_h, p_h, v_h)$.
• Numerical Traces: The method introduces numerical traces on element boundaries ($\hat{u}_h, \hat{q}_h, \hat{p}_h, \hat{v}_h$) to weakly enforce continuity.
• Stabilization: Carefully designed stabilization parameters ($\tau_{pu}, \tau_{vq}, \tau_{qv}, \tau_f$) are introduced in the numerical fluxes to ensure stability. Specifically, $\tau_f$ is chosen to control the nonlinear flux $f(u)$.
• Static Condensation: Interior degrees of freedom are eliminated locally, resulting in a global system that only involves the trace unknowns on the mesh skeleton. This significantly reduces the computational cost compared to standard DG.

C. Time Discretization

The semi-discrete scheme is coupled with an implicit $\theta$-method ($\theta \in [1/2, 1]$) for time stepping:

• $\theta = 1/2$: Crank-Nicolson (second-order accurate, energy conservative).
• $\theta = 1$: Backward Euler (first-order accurate, unconditionally stable with numerical dissipation).

3. Key Contributions

1. First HDG for Full Ostrovsky Equation: This is the first proposal of an HDG method for the full Ostrovsky equation (including both dispersion and the nonlocal Coriolis term).
2. Localization Strategy: A novel mixed formulation that localizes the nonlocal operator $\partial_x^{-1}$ via an auxiliary variable $v$ with a specific boundary constraint, making it amenable to standard HDG frameworks.
3. Stability Analysis:
   • Proved $L^2$-stability for the semi-discrete and fully discrete schemes under specific conditions on the stabilization parameters.
   • Demonstrated that for periodic problems with $\beta > 0, \gamma > 0$, the scheme can be made exactly energy-conservative by choosing specific parameter values (e.g., $\tau_{vq} = \sqrt{\beta/\gamma}$).
4. Error Analysis:
   • Derived a priori $L^2$-error estimates for the smooth solution $u$.
   • Proved a convergence rate of $O(h^{k+1/2})$ in the $L^2$-norm, explicitly accounting for the nonlinear convective flux and the nonlocal term.
   • Established time-discretization error bounds showing $O(\Delta t)$ for $\theta \neq 1/2$ and $O(\Delta t^2)$ for $\theta = 1/2$.
5. Handling Singular Limits: The method is shown to robustly handle the singular limit $\beta \to 0$ (transition to the Ostrovsky-Hunter equation) and capture non-smooth "peakon" solutions.

4. Results and Numerical Experiments

The authors validated the theory through several numerical examples:

• Convergence Tests: Using a smooth manufactured solution, the method demonstrated optimal convergence rates. While the theoretical bound is $O(h^{k+1/2})$, the observed rates were often closer to $O(h^{k+1})$, indicating superconvergence typical of hybridized methods.
• Solitary Wave Propagation: The scheme successfully propagated solitary waves on a periodic domain without spurious oscillations or significant shape distortion over long time intervals, preserving the coherent structure.
• Peakon and Singular Limit ($\beta \to 0$):
  • The method accurately simulated the peakon solution (a solution with a sharp corner) of the Ostrovsky-Hunter equation ($\beta=0$).
  • As $\beta$ decreased ($10^{-4} \to 10^{-6}$), the numerical solutions converged to the exact OH solution, demonstrating the method's capability to handle the singular limit and non-smooth profiles.

5. Significance

• Computational Efficiency: By reducing the global system size via static condensation, the HDG method offers a more efficient alternative to standard DG methods for high-order dispersive problems.
• Robustness: The method handles the complex interplay of nonlinearity, high-order dispersion, and nonlocal operators simultaneously, a challenge where standard finite difference or spectral methods often struggle with stability or spurious oscillations.
• Physical Relevance: The ability to model both smooth solitary waves and non-smooth peakons, as well as the transition between dispersive and non-dispersive regimes, makes this method highly relevant for oceanographic modeling of internal waves under the Coriolis effect.
• Extensibility: The authors note that this framework naturally extends to two-dimensional KP-type models involving transverse nonlocal terms.

In summary, this paper provides a rigorous mathematical foundation and a robust numerical framework for solving the Ostrovsky equation, overcoming the difficulties posed by its nonlocal and high-order nature through a tailored HDG approach.