A hybridizable discontinuous Galerkin method for the Ostrovsky equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910022296928256 |
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| author | Dwivedi, Mukul Rupp, Andreas |
| author_facet | Dwivedi, Mukul Rupp, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13786 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A hybridizable discontinuous Galerkin method for the Ostrovsky equation Dwivedi, Mukul Rupp, Andreas Numerical Analysis 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. |
| title | A hybridizable discontinuous Galerkin method for the Ostrovsky equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.13786 |