Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

Abstract

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.

Authors:
ORCiD logo [1];  [2];  [3];  [4]
+ Show Author Affiliations
  1. Univ. of Nebraska, Lincoln, NE (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Texas A & M Univ., College Station, TX (United States); Weizmann Inst. of Science, Rehovot (Israel)
  4. Univ. of Exeter (United Kingdom)
Publication Date:
Research Org.:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1357141
Report Number(s):
LA-UR-17-23192
Journal ID: ISSN 0935-4964
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Theoretical and Computational Fluid Dynamics
Additional Journal Information:
Journal Volume: 32; Journal Issue: 1; Journal ID: ISSN 0935-4964
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics; Euler-Voigt; Navier-Stokes-Voigt; Inviscid Regularization, Turbulence Models; Inviscid Regularization; Turbulence Models

Citation Formats

Larios, Adam, Petersen, Mark R., Titi, Edriss S., and Wingate, Beth. A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization. United States: N. p., 2017. Web. doi:10.1007/s00162-017-0434-0.
Copy to clipboard
Larios, Adam, Petersen, Mark R., Titi, Edriss S., & Wingate, Beth. A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization. United States. https://doi.org/10.1007/s00162-017-0434-0
Copy to clipboard
Larios, Adam, Petersen, Mark R., Titi, Edriss S., and Wingate, Beth. Sat . "A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization". United States. https://doi.org/10.1007/s00162-017-0434-0. https://www.osti.gov/servlets/purl/1357141.
Copy to clipboard
@article{osti_1357141,
title = {A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization},
author = {Larios, Adam and Petersen, Mark R. and Titi, Edriss S. and Wingate, Beth},
abstractNote = {We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.},
doi = {10.1007/s00162-017-0434-0},
journal = {Theoretical and Computational Fluid Dynamics},
number = 1,
volume = 32,
place = {United States},
year = {Sat Apr 29 00:00:00 EDT 2017},
month = {Sat Apr 29 00:00:00 EDT 2017}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1007/s00162-017-0434-0
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Citation Metrics:
Cited by: 17 works
Citation information provided by
Web of Science
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Geometric Properties and Nonblowup of 3D Incompressible Euler Flow
journal, April 2005

  • Deng, Jian; Hou, Thomas Y.; Yu, Xinwei
  • Communications in Partial Differential Equations, Vol. 30, Issue 1-2
  • DOI: 10.1081/PDE-200044488

Energy conservation and Onsager's conjecture for the Euler equations
journal, April 2008

Evidence for a singularity of the three‐dimensional, incompressible Euler equations
journal, July 1993

  • Kerr, Robert M.
  • Physics of Fluids A: Fluid Dynamics, Vol. 5, Issue 7
  • DOI: 10.1063/1.858849

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations
journal, September 2009

  • Kalantarov, Varga K.; Titi, Edriss S.
  • Chinese Annals of Mathematics, Series B, Vol. 30, Issue 6
  • DOI: 10.1007/s11401-009-0205-3

Viscosity versus vorticity stretching: Global well-posedness for a family of Navier–Stokes-alpha-like models
journal, June 2007

  • Olson, Eric; Titi, Edriss S.
  • Nonlinear Analysis: Theory, Methods & Applications, Vol. 66, Issue 11
  • DOI: 10.1016/j.na.2006.03.030

Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer
journal, November 1994

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations
journal, March 2002

  • Kozono, Hideo; Ogawa, Takayoshi; Taniuchi, Yasushi
  • Mathematische Zeitschrift, Vol. 242, Issue 2
  • DOI: 10.1007/s002090100332

Remarks on a paper by J. T. Beale, T. Kato, and A. Majda
journal, September 1985

  • Ponce, Gustavo
  • Communications in Mathematical Physics, Vol. 98, Issue 3
  • DOI: 10.1007/BF01205787

Onsager's conjecture on the energy conservation for solutions of Euler's equation
journal, October 1994

  • Constantin, Peter; Weinan, E.; Titi, Edriss S.
  • Communications in Mathematical Physics, Vol. 165, Issue 1
  • DOI: 10.1007/BF02099744

Dissipative Euler flows and Onsager's conjecture
journal, January 2014

  • De Lellis, Camillo; Székelyhidi Jr., László
  • Journal of the European Mathematical Society, Vol. 16, Issue 7
  • DOI: 10.4171/JEMS/466

Numerical solution of the Navier-Stokes equations
journal, January 1968

On relaxation times in the Navier-Stokes-Voigt model
journal, March 2013

  • Layton, William J.; Rebholz, Leo G.
  • International Journal of Computational Fluid Dynamics, Vol. 27, Issue 3
  • DOI: 10.1080/10618562.2013.766328

3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis
journal, June 2011

Numerical simulations of possible finite time singularities in the incompressible Euler equations: Comparison of numerical methods
journal, August 2008

  • Grafke, Tobias; Homann, Holger; Dreher, Jürgen
  • Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17
  • DOI: 10.1016/j.physd.2007.11.006

Remarks on the breakdown of smooth solutions for the 3-D Euler equations
journal, March 1984

  • Beale, J. T.; Kato, T.; Majda, A.
  • Communications in Mathematical Physics, Vol. 94, Issue 1
  • DOI: 10.1007/BF01212349

Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models
journal, January 2006

  • Cao, Yanping; Lunasin, Evelyn M.; Titi, Edriss S.
  • Communications in Mathematical Sciences, Vol. 4, Issue 4
  • DOI: 10.4310/CMS.2006.v4.n4.a8

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations
journal, January 2014

  • Hou, Thomas Y.; Lei, Zhen; Luo, Guo
  • Archive for Rational Mechanics and Analysis, Vol. 212, Issue 2
  • DOI: 10.1007/s00205-013-0717-6

On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model
journal, January 2010

  • Levant, Boris; Ramos, Fabio; Titi, Edriss S.
  • Communications in Mathematical Sciences, Vol. 8, Issue 1
  • DOI: 10.4310/CMS.2010.v8.n1.a14

On NAVIER-STOKES and KELVIN-VOIGT Equations in Three Dimensions in Interpolation Spaces
journal, January 1992

<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1993-01-01">January 1993</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Constantin, Peter; Fefferman, Charles</span> </li> <li> Indiana University Mathematics Journal, Vol. 42, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1512/iumj.1993.42.42034" class="text-muted" target="_blank" rel="noopener noreferrer">10.1512/iumj.1993.42.42034<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/RevModPhys.78.87" target="_blank" rel="noopener noreferrer" class="name">Onsager and the theory of hydrodynamic turbulence<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2006-01-01">January 2006</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Eyink, Gregory L.; Sreenivasan, Katepalli R.</span> </li> <li> Reviews of Modern Physics, Vol. 78, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/RevModPhys.78.87" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/RevModPhys.78.87<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.aml.2012.12.004" target="_blank" rel="noopener noreferrer" class="name">Finite-time Euler singularities: A Lagrangian perspective<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-04-01">April 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Grafke, Tobias; Grauer, Rainer</span> </li> <li> Applied Mathematics Letters, Vol. 26, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.aml.2012.12.004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.aml.2012.12.004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/BF00247678" target="_blank" rel="noopener noreferrer" class="name">Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I)<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1969-01-01">January 1969</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Témam, R.</span> </li> <li> Archive for Rational Mechanics and Analysis, Vol. 32, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/BF00247678" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/BF00247678<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.physd.2008.01.018" target="_blank" rel="noopener noreferrer" class="name">Blowup or no blowup? The interplay between theory and numerics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2008-08-01">August 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Hou, Thomas Y.; Li, Ruo</span> </li> <li> Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.physd.2008.01.018" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.physd.2008.01.018<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/BF02097394" target="_blank" rel="noopener noreferrer" class="name">On the blow-up of solutions of the 3-D Euler equations in a bounded domain<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1993-07-01">July 1993</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Ferrari, Andrew B.</span> </li> <li> Communications in Mathematical Physics, Vol. 155, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/BF02097394" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/BF02097394<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevE.86.066302" target="_blank" rel="noopener noreferrer" class="name">Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-12-01">December 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bustamante, Miguel D.; Brachet, Marc</span> </li> <li> Physical Review E, Vol. 86, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevE.86.066302" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevE.86.066302<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.physd.2007.10.014" target="_blank" rel="noopener noreferrer" class="name">The three-dimensional Euler equations: Where do we stand?<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2008-08-01">August 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Gibbon, J. D.</span> </li> <li> Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.physd.2007.10.014" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.physd.2007.10.014<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00021-014-0167-4" target="_blank" rel="noopener noreferrer" class="name">A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-02-20">February 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Cheskidov, A.; Shvydkoy, R.</span> </li> <li> Journal of Mathematical Fluid Mechanics, Vol. 16, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00021-014-0167-4" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00021-014-0167-4<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.fluiddyn.2004.09.005" target="_blank" rel="noopener noreferrer" class="name">Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2005-04-01">April 2005</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Cichowlas, C.; Brachet, M-E</span> </li> <li> Fluid Dynamics Research, Vol. 36, Issue 4-6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.fluiddyn.2004.09.005" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.fluiddyn.2004.09.005<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00332-008-9029-7" target="_blank" rel="noopener noreferrer" class="name">Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2008-11-06">November 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Kalantarov, Varga K.; Levant, Boris; Titi, Edriss S.</span> </li> <li> Journal of Nonlinear Science, Vol. 19, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00332-008-9029-7" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00332-008-9029-7<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.camwa.2012.07.010" target="_blank" rel="noopener noreferrer" class="name">Numerical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-10-01">October 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Kuberry, Paul; Larios, Adam; Rebholz, Leo G.</span> </li> <li> Computers & Mathematics with Applications, Vol. 64, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.camwa.2012.07.010" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.camwa.2012.07.010<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.3934/dcdss.2010.3.185" target="_blank" rel="noopener noreferrer" class="name">Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-04-01">April 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Titi, E.; Bardos, Claude</span> </li> <li> Discrete and Continuous Dynamical Systems - Series S, Vol. 3, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.3934/dcdss.2010.3.185" class="text-muted" target="_blank" rel="noopener noreferrer">10.3934/dcdss.2010.3.185<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1017/S0022112083001159" target="_blank" rel="noopener noreferrer" class="name">Small-scale structure of the Taylor–Green vortex<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1983-05-01">May 1983</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Brachet, Marc E.; Meiron, Daniel I.; Orszag, Steven A.</span> </li> <li> Journal of Fluid Mechanics, Vol. 130, Issue -1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1017/S0022112083001159" class="text-muted" target="_blank" rel="noopener noreferrer">10.1017/S0022112083001159<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00332-013-9175-4" target="_blank" rel="noopener noreferrer" class="name">The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-06-22">June 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Gibbon, John D.; Titi, Edriss S.</span> </li> <li> Journal of Nonlinear Science, Vol. 23, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00332-013-9175-4" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00332-013-9175-4<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.4007/annals.2015.182.1.3" target="_blank" rel="noopener noreferrer" class="name">Anomalous dissipation for 1/5-Hölder Euler flows<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-07-01">July 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Buckmaster, Tristan; De Lellis, Camillo; Isett, Philip</span> </li> <li> Annals of Mathematics</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.4007/annals.2015.182.1.3" class="text-muted" target="_blank" rel="noopener noreferrer">10.4007/annals.2015.182.1.3<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1098/rsta.1972.0032" target="_blank" rel="noopener noreferrer" class="name">Model Equations for Long Waves in Nonlinear Dispersive Systems<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1972-03-30">March 1972</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Benjamin, T. B.; Bona, J. L.; Mahony, J. J.</span> </li> <li> Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 272, Issue 1220</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1098/rsta.1972.0032" class="text-muted" target="_blank" rel="noopener noreferrer">10.1098/rsta.1972.0032<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1093/imamat/hxr069" target="_blank" rel="noopener noreferrer" class="name">The Navier-Stokes-Voight model for image inpainting<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-03-20">March 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Ebrahimi, M. A.; Lunasin, E.</span> </li> <li> IMA Journal of Applied Mathematics, Vol. 78, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1093/imamat/hxr069" class="text-muted" target="_blank" rel="noopener noreferrer">10.1093/imamat/hxr069<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s11565-009-0069-1" target="_blank" rel="noopener noreferrer" class="name">Global existence for a regularized magnetohydrodynamic-α model<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2009-06-03">June 2009</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Catania, Davide</span> </li> <li> ANNALI DELL'UNIVERSITA' DI FERRARA, Vol. 56, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s11565-009-0069-1" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s11565-009-0069-1<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00205-008-0201-x" target="_blank" rel="noopener noreferrer" class="name">On Admissibility Criteria for Weak Solutions of the Euler Equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2008-12-02">December 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> de Lellis, Camillo; Székelyhidi, László</span> </li> <li> Archive for Rational Mechanics and Analysis, Vol. 195, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00205-008-0201-x" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00205-008-0201-x<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1137/140966411" target="_blank" rel="noopener noreferrer" class="name">Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-01-01">January 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Luo, Guo; Hou, Thomas Y.</span> </li> <li> Multiscale Modeling & Simulation, Vol. 12, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1137/140966411" class="text-muted" target="_blank" rel="noopener noreferrer">10.1137/140966411<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevE.92.013020" target="_blank" rel="noopener noreferrer" class="name">Self-truncation and scaling in Euler-Voigt- <math> <mi>α</mi> </math> and related fluid models<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-07-01">July 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Di Molfetta, Giuseppe; Krstlulovic, Giorgio; Brachet, Marc</span> </li> <li> Physical Review E, Vol. 92, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevE.92.013020" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevE.92.013020<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.3934/dcds.2010.28.375" target="_blank" rel="noopener noreferrer" class="name">Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-04-01">April 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Ramos, Fabio; Titi, Edriss</span> </li> <li> Discrete and Continuous Dynamical Systems, Vol. 28, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.3934/dcds.2010.28.375" class="text-muted" target="_blank" rel="noopener noreferrer">10.3934/dcds.2010.28.375<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.physd.2009.09.016" target="_blank" rel="noopener noreferrer" class="name">Calculation of complex singular solutions to the 3D incompressible Euler equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2009-12-01">December 2009</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Siegel, M.; Caflisch, R. E.</span> </li> <li> Physica D: Nonlinear Phenomena, Vol. 238, Issue 23-24</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.physd.2009.09.016" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.physd.2009.09.016<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00220-014-2262-z" target="_blank" rel="noopener noreferrer" class="name">Onsager’s Conjecture Almost Everywhere in Time<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-01-07">January 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Buckmaster, Tristan</span> </li> <li> Communications in Mathematical Physics, Vol. 333, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00220-014-2262-z" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00220-014-2262-z<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00222-012-0429-9" target="_blank" rel="noopener noreferrer" class="name">Dissipative continuous Euler flows<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-10-25">October 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> De Lellis, Camillo; Székelyhidi, László</span> </li> <li> Inventiones mathematicae, Vol. 193, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00222-012-0429-9" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00222-012-0429-9<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/BF01009458" target="_blank" rel="noopener noreferrer" class="name">The Taylor-Green vortex and fully developed turbulence<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1984-03-01">March 1984</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Brachet, M. E.; Meiron, D.; Orszag, S.</span> </li> <li> Journal of Statistical Physics, Vol. 34, Issue 5-6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/BF01009458" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/BF01009458<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1080/03091920903304080" target="_blank" rel="noopener noreferrer" class="name">The dynamics of unforced turbulence at high Reynolds number for Taylor–Green vortices generalized to MHD<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-04-01">April 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Pouquet, A.; Lee, E.; Brachet, M. E.</span> </li> <li> Geophysical & Astrophysical Fluid Dynamics, Vol. 104, Issue 2-3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1080/03091920903304080" class="text-muted" target="_blank" rel="noopener noreferrer">10.1080/03091920903304080<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s00332-006-0800-3" target="_blank" rel="noopener noreferrer" class="name">Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2006-08-22">August 2006</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Hou, Thomas Y.; Li, Ruo</span> </li> <li> Journal of Nonlinear Science, Vol. 16, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s00332-006-0800-3" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s00332-006-0800-3<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1002/cpa.20218" target="_blank" rel="noopener noreferrer" class="name">An inviscid regularization for the surface quasi‐geostrophic equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2008-10-01">October 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Khouider, Boualem; Titi, Edriss S.</span> </li> <li> Communications on Pure and Applied Mathematics, Vol. 61, Issue 10</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1002/cpa.20218" class="text-muted" target="_blank" rel="noopener noreferrer">10.1002/cpa.20218<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="sr-only">Previous Page</span><span class="fa fa-angle-left"></span></a> <ul class="pagination d-inline-block" style="padding-left:.2em;"></ul> <a class="pure-button next page" href="#" rel="next"><span class="sr-only">Next Page</span><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="journal"><span class="fa fa-angle-right"></span> journal<small class="text-muted"> (50)</small></a></li> </ul> <div style="margin-top:2em;"> <form class="pure-form small text-muted reference-search"> <label for="reference-search-text" class="sr-only">Search</label> <input class="search form-control pure-input-1" id="reference-search-text" placeholder="Search" style="margin-bottom:10px;" /> <fieldset aria-label="Sort By"> <legend class="legend-filters sr-only">Sort by:</legend> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="name" style="position:relative;top:2px;" id="reference-search-sort-name"><label for="reference-search-sort-name" style="margin-left: .3em;">Sort by title</label></div> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="date" data-order="desc" style="position:relative;top:2px;" id="reference-search-sort-date"><label for="reference-search-sort-date" style="margin-left: .3em;">Sort by date</label></div> </fieldset> <div class="text-left" style="margin-left:1em;"> <a href="" class="filter-clear clearfix" title="Clear filter / sort" style="font-weight:normal; float:none;">[ × clear filter / sort ]</a> </div> <input type="submit" id="sort_submit_references" name="submit" aria-label="submit" style="display: none;"/> </form> </div> </div> </div> </section> <section id="biblio-related" class="tab-content tab-content-sec " data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <section id="biblio-similar" class="tab-content tab-content-sec active" data-tab="related"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="0" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/441146-geometric-constraints-potentially-singular-solutions-euler-equations" itemprop="url">Geometric constraints on potentially singular solutions for the 3-D Euler equations</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Constantin, P</span> ; <span class="author">Fefferman, C</span> ; <span class="author">Majda, A J</span> <span class="text-muted pubdata"> - Communications in Partial Differential Equations</span> </span> </div> <div class="abstract">We discuss necessary and sufficient conditions for the formation of finite time singularities (blow up) in the incompressible three dimensional Euler equations. The well-known result of Beale, Kato and Majda states that these equations have smooth solutions on the time interval (0,t) if, and only if lim/t{r_arrow}T {integral}{sup t}{sub 0} {parallel}{Omega}({center_dot},s){parallel}{sub L}{sup {infinity}} (dx)dx < {infinity} where {Omega} = {triangledown} X u is the vorticity of the fluid and u is its divergence=free velocity. In this paper we prove criteria in which the direction of vorticity {xi} = {Omega}/{vert_bar}{Omega}{vert_bar} plays an important role.</div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/10166132-adaptive-projection-method-incompressible-euler-equations" itemprop="url">An adaptive projection method for the incompressible Euler equations</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Conference</small><span class="authors"> <span class="author">Almgren, A S</span> ; <span class="author">Bell, J B</span> ; <span class="author">Howell, L H</span> ; <span class="author">...</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">In this paper we present a method for solving the time-dependent incompressible Euler equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the convection step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid flow. Our approach to adaptive refinement uses a nested hierarchy of grids with simultaneous refinement of the grids in both space<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> and time. The integration algorithm on the grid hierarchy is a recursive procedure in which a coarse grid is advanced, fine grids are advanced multiple steps to reach the same time as the coarse grid and the grids are then synchronized. We will describe the integration algorithm in detail, with emphasis on the projection used to enforce the incompressibility constraint. Numerical examples are presented to demonstrate the convergence properties of the method and to illustrate the behavior of the method at the interface between coarse and fine grids. An additional example demonstrates the performance of the method on a more realistic problem.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/6127044-adaptive-projection-method-incompressible-euler-equations" itemprop="url">An adaptive projection method for the incompressible Euler equations</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Conference</small><span class="authors"> <span class="author">Almgren, A S</span> ; <span class="author">Bell, J B</span> ; <span class="author">Howell, L H</span> ; <span class="author">...</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">In this paper we present a method for solving the time-dependent incompressible Euler equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the convection step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid flow. Our approach to adaptive refinement uses a nested hierarchy of grids with simultaneous refinement of the grids in both space<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> and time. The integration algorithm on the grid hierarchy is a recursive procedure in which a coarse grid is advanced, fine grids are advanced multiple steps to reach the same time as the coarse grid and the grids are then synchronized. We will describe the integration algorithm in detail, with emphasis on the projection used to enforce the incompressibility constraint. Numerical examples are presented to demonstrate the convergence properties of the method and to illustrate the behavior of the method at the interface between coarse and fine grids. An additional example demonstrates the performance of the method on a more realistic problem.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1827275-optimal-renormalization-multiscale-systems" itemprop="url">Optimal renormalization of multiscale systems</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Price, Jacob</span> ; <span class="author">Meuris, Brek</span> ; <span class="author">Shapiro, Madelyn R.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Proceedings of the National Academy of Sciences of the United States of America</span> </span> </div> <div class="abstract">While model order reduction is a promising approach in dealing with multi-scale time-dependent systems that are too large or too expensive to simulate for long times, the resulting reduced order models can suffer from instabilities. We have recently developed a time-dependent renormalization approach to stabilize such reduced models. In the current work, we extend this framework by introducing a parameter that controls the time-decay of the memory of such models and optimally selecting this parameter based on limited fully resolved simulations. First, we demonstrate our framework on the inviscid Burgers equation whose solution develops a finite-time singularity. Our renormalized reduced<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> order models are stable and accurate for long times while using for their calibration only data from a full order simulation before the occurrence of the singularity. Furthermore, we apply this framework to the 3D Euler equations of incompressible fluid flow, where the problem of finite-time singularity formation is still open and where brute force simulation is only feasible for short times. Our approach allows us to obtain for the first time a perturbatively renormalizable model which is stable for long times and includes all the complex effects present in the 3D Euler dynamics. We find that, in each application, the renormalization coefficients display algebraic decay with increasing resolution, and that the parameter which controls the time-decay of the memory is problem-dependent.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1073/pnas.2102266118" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1827275" data-product-type="Journal Article" data-product-subtype="AC" >https://doi.org/10.1073/pnas.2102266118</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/198204-mesh-adaptive-collocation-technique-simulation-advection-dominated-single-multiphase-transport-phenomena-porous-media" itemprop="url">A mesh-adaptive collocation technique for the simulation of advection-dominated single- and multiphase transport phenomena in porous media</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Conference</small><span class="authors"> <span class="author">Koch, M</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">A new mesh-adaptive 1D collocation technique has been developed to efficiently solve transient advection-dominated transport problems in porous media that are governed by a hyperbolic/parabolic (singularly perturbed) PDE. After spatial discretization a singularly perturbed ODE is obtained which is solved by a modification of the COLNEW ODE-collocation code. The latter also contains an adaptive mesh procedure that has been enhanced here to resolve linear and nonlinear transport flow problems with steep fronts where regular FD and FE methods often fail. An implicit first-order backward Euler and a third-order Taylor-Donea technique are employed for the time integration. Numerical simulations on a<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> variety of high Peclet-number transport phenomena as they occur in realistic porous media flow situations are presented. Examples include classical linear advection-diffusion, nonlinear adsorption, two-phase Buckley-Leverett flow without and with capillary forces (Rapoport-Leas equation) and Burgers` equation for inviscid fluid flow. In most of these examples sharp fronts and/or shocks develop which are resolved in an oscillation-free manner by the present adaptive collocation method. The backward Euler method has some amount of numerical dissipation is observed when the time-steps are too large. The third-order Taylor-Donea technique is less dissipative but is more prone to numerical oscillations. The simulations show that for the efficient solution of nonlinear singularly perturbed PDE`s governing flow transport a careful balance must be struck between the optimal mesh adaptation, the nonlinear iteration method and the time-stepping procedure. More theoretical research is needed with this regard.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9;"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator"/> <br/> <div class="col text-center mt-3"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.osti.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="col text-center small" style="margin-top: 0.5em;margin-bottom:2.0rem;"> <div class="row justify-content-center" style="color:white"> <div class="pure-menu pure-menu-horizontal" style='white-space:normal'> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="https://www.osti.gov/disclaim" class="pure-menu-link" target="_blank" ref="noopener noreferrer"><span class="fa fa-institution"></span> Website Policies <span class="d-none d-sm-inline d-print-none" style="color:#737373;">/ Important Links</span></a></li> <li class="pure-menu-item" style='float:none;'><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span>Contact Us</a></li> <li class="d-block d-md-none mb-1"></li> <li class="pure-menu-item" style='float:none;'><a target="_blank" title="Vulnerability Disclosure Program" class="pure-menu-link" href="https://doe.responsibledisclosure.com/hc/en-us" rel="noopener noreferrer">Vulnerability Disclosure Program</a></li> <li class="d-block d-lg-none mb-1"></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.facebook.com/ostigov" target="_blank" class="pure-menu-link social ext fa fa-facebook" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Facebook</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://twitter.com/OSTIgov" target="_blank" class="pure-menu-link social ext fa fa-twitter" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Twitter</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.youtube.com/user/ostigov" target="_blank" class="pure-menu-link social ext fa fa-youtube-play" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Youtube</span></a></li> </ul> </div> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.240327.0205.css" rel="stylesheet"> <script src="/pages/js/pages.240327.0205.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.240327.0205.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript> <script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript> </body> <!-- DOE PAGES v.240327.0205 --> </html>