Solutions of a nonhyperbolic pair of balance laws

Michael Sever

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 1, page 37-58
  • ISSN: 0764-583X

Abstract

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We describe a constructive algorithm for obtaining smooth solutions of a nonlinear, nonhyperbolic pair of balance laws modeling incompressible two-phase flow in one space dimension and time. Solutions are found as stationary solutions of a related hyperbolic system, based on the introduction of an artificial time variable. As may be expected for such nonhyperbolic systems, in general the solutions obtained do not satisfy both components of the given initial data. This deficiency may be overcome, however, by introducing an alternative “solution" satisfying both components of the initial data and an approximate form of a corresponding linearized system.

How to cite

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Sever, Michael. "Solutions of a nonhyperbolic pair of balance laws." ESAIM: Mathematical Modelling and Numerical Analysis 39.1 (2010): 37-58. <http://eudml.org/doc/194257>.

@article{Sever2010,
abstract = { We describe a constructive algorithm for obtaining smooth solutions of a nonlinear, nonhyperbolic pair of balance laws modeling incompressible two-phase flow in one space dimension and time. Solutions are found as stationary solutions of a related hyperbolic system, based on the introduction of an artificial time variable. As may be expected for such nonhyperbolic systems, in general the solutions obtained do not satisfy both components of the given initial data. This deficiency may be overcome, however, by introducing an alternative “solution" satisfying both components of the initial data and an approximate form of a corresponding linearized system. },
author = {Sever, Michael},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonhyperbolic balance laws; incompressible two-fluid flow.; algorithm; smooth solutions; incompressible two-phase flow; hyperbolic system},
language = {eng},
month = {3},
number = {1},
pages = {37-58},
publisher = {EDP Sciences},
title = {Solutions of a nonhyperbolic pair of balance laws},
url = {http://eudml.org/doc/194257},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Sever, Michael
TI - Solutions of a nonhyperbolic pair of balance laws
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 1
SP - 37
EP - 58
AB - We describe a constructive algorithm for obtaining smooth solutions of a nonlinear, nonhyperbolic pair of balance laws modeling incompressible two-phase flow in one space dimension and time. Solutions are found as stationary solutions of a related hyperbolic system, based on the introduction of an artificial time variable. As may be expected for such nonhyperbolic systems, in general the solutions obtained do not satisfy both components of the given initial data. This deficiency may be overcome, however, by introducing an alternative “solution" satisfying both components of the initial data and an approximate form of a corresponding linearized system.
LA - eng
KW - Nonhyperbolic balance laws; incompressible two-fluid flow.; algorithm; smooth solutions; incompressible two-phase flow; hyperbolic system
UR - http://eudml.org/doc/194257
ER -

References

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  8. B.L. Keyfitz, R. Sanders and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow . Discrete Contin. Dynam. Systems, Series B3 (2003) 541–563.  Zbl1055.35074
  9. B.L. Keyfitz, M. Sever and F. Zhang, Viscous singular shock structure for a nonhyperbolic two-fluid model. Nonlinearity17 (2004) 1731–1747.  Zbl1077.35091
  10. H.-O. Kreiss and J. Ystrom, Parabolic problems which are ill-posed in the zero dissipation limit . Math. Comput. Model.35 (2002) 1271–1295.  Zbl1066.76064
  11. M.S. Mock, Systems of conservation laws of mixed type . J. Diff. Equations37 (1980) 70–88.  Zbl0413.34017
  12. H. Ransom and D.L. Hicks, Hyperbolic two-pressure models for two-phase flow . J. Comput. Phys.53 (1984) 124–151.  Zbl0537.76070
  13. R. Sanders and M. Sever, Computations with singular shocks (2005) (preprint).  
  14. S. Sever, A model of discontinuous, incompressible two-phase flow (2005) (preprint).  
  15. H.B. Stewart and B. Wendroff, Two-phase flow: models and methods . J. Comput. Phys.56 (1984) 363–409.  Zbl0596.76103

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