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Hydrodynamics of a discrete conservation law.

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Title: Hydrodynamics of a discrete conservation law.
Authors: Sprenger, Patrick1 (AUTHOR) sprenger@ucmerced.edu, Chong, Christopher2 (AUTHOR), Okyere, Emmanuel2 (AUTHOR), Herrmann, Michael3 (AUTHOR), Kevrekidis, P. G.4 (AUTHOR), Hoefer, Mark A.5 (AUTHOR)
Source: Studies in Applied Mathematics. Nov2024, Vol. 153 Issue 4, p1-48. 48p.
Subjects: Modulation theory, Riemann-Hilbert problems, Shock waves, Conservation laws (Physics), Hydrodynamics
Abstract: The Riemann problem for the discrete conservation law 2u̇n+un+12−un−12=0$2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ 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 analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup 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 counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an 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. [ABSTRACT FROM AUTHOR]
Copyright of Studies in Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Hydrodynamics of a discrete conservation law.
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  Data: <searchLink fieldCode="AR" term="%22Sprenger%2C+Patrick%22">Sprenger, Patrick</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> sprenger@ucmerced.edu</i><br /><searchLink fieldCode="AR" term="%22Chong%2C+Christopher%22">Chong, Christopher</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Okyere%2C+Emmanuel%22">Okyere, Emmanuel</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Herrmann%2C+Michael%22">Herrmann, Michael</searchLink><relatesTo>3</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kevrekidis%2C+P%2E+G%2E%22">Kevrekidis, P. G.</searchLink><relatesTo>4</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Hoefer%2C+Mark+A%2E%22">Hoefer, Mark A.</searchLink><relatesTo>5</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Studies+in+Applied+Mathematics%22">Studies in Applied Mathematics</searchLink>. Nov2024, Vol. 153 Issue 4, p1-48. 48p.
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  Data: <searchLink fieldCode="DE" term="%22Modulation+theory%22">Modulation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Riemann-Hilbert+problems%22">Riemann-Hilbert problems</searchLink><br /><searchLink fieldCode="DE" term="%22Shock+waves%22">Shock waves</searchLink><br /><searchLink fieldCode="DE" term="%22Conservation+laws+%28Physics%29%22">Conservation laws (Physics)</searchLink><br /><searchLink fieldCode="DE" term="%22Hydrodynamics%22">Hydrodynamics</searchLink>
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  Data: The Riemann problem for the discrete conservation law 2u̇n+un+12−un−12=0$2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ 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 analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup 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 counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Studies in Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1111/sapm.12767
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        Text: English
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        Type: general
      – SubjectFull: Riemann-Hilbert problems
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      – SubjectFull: Shock waves
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      – SubjectFull: Conservation laws (Physics)
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              M: 11
              Text: Nov2024
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