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Solutions of a nonhyperbolic pair of balance laws

Published online by Cambridge University Press:  15 March 2005

Michael Sever
Michael Sever*
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel. server@math.huji.ac.il
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Abstract

We describe a constructive algorithm for obtaining smoothsolutions of a nonlinear, nonhyperbolic pair of balance lawsmodeling incompressible two-phase flow in one space dimension andtime. Solutions are found as stationary solutions of a relatedhyperbolic system, based on the introduction of an artificial timevariable.As may be expected for such nonhyperbolic systems, in general thesolutions obtained do not satisfy both components of the giveninitial data. This deficiency may be overcome, however, byintroducing an alternative “solution" satisfying both componentsof the initial data and an approximate form of a correspondinglinearized system.

Keywords

Nonhyperbolic balance lawsincompressible two-fluid flow.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 37 - 58
Copyright
© EDP Sciences, SMAI, 2005

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References

Amadori, D., Gosse, L. and Guerra, G., Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162 (2002) 327–366. CrossRef
A. Bressan, Hyperbolic Systems of Conservation Laws: the One-dimensional Cauchy Problem. Oxford University Press (2000).
DiPerna, R.J., Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28 (1979) 202–212. CrossRef
T.N. Dinh, R.R. Nourgaliev and T.G. Theofanous, Understanding the ill-posed two-fluid model , in Proc. of the 10th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-10) Seoul, Korea (October 2003).
D.A. Drew and S.L. Passman, Theory of Multicomponent Fluids . Springer, New York (1999).
Godunov, S.K., An interesting class of quasilinear systems. Dokl. Akad. Nauk SSR 139 (1961) 521–523.
M. Ishii, Thermo-fluid dynamic theory of two-phase flow . Eyrolles, Paris (1975).
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 B 3 (2003) 541–563.
Keyfitz, B.L., Sever, M. and Zhang, F., Viscous singular shock structure for a nonhyperbolic two-fluid model. Nonlinearity 17 (2004) 1731–1747. CrossRef
Kreiss, H.-O. and Ystrom, J., Parabolic problems which are ill-posed in the zero dissipation limit . Math. Comput. Model. 35 (2002) 1271–1295. CrossRef
Mock, M.S., Systems of conservation laws of mixed type . J. Diff. Equations 37 (1980) 70–88. CrossRef
Ransom, H. and Hicks, D.L., Hyperbolic two-pressure models for two-phase flow . J. Comput. Phys. 53 (1984) 124–151. CrossRef
R. Sanders and M. Sever, Computations with singular shocks (2005) (preprint).
S. Sever, A model of discontinuous, incompressible two-phase flow (2005) (preprint).
Stewart, H.B. and Wendroff, B., Two-phase flow: models and methods . J. Comput. Phys. 56 (1984) 363–409. CrossRef