Hydrodynamics of a Discrete Conservation Law
Fuente:
arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915294130208768 |
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| author | Sprenger, Patrick Chong, Christopher Okyere, Emmanuel Herrmann, Michael Kevrekidis, P. G. Hoefer, Mark A. |
| author_facet | Sprenger, Patrick Chong, Christopher Okyere, Emmanuel Herrmann, Michael Kevrekidis, P. G. Hoefer, Mark A. |
| contents | The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hydrodynamics of a Discrete Conservation Law Sprenger, Patrick Chong, Christopher Okyere, Emmanuel Herrmann, Michael Kevrekidis, P. G. Hoefer, Mark A. Pattern Formation and Solitons The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. |
| title | Hydrodynamics of a Discrete Conservation Law |
| topic | Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2404.16750 |