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.

Keywords: nonhyperbolic balance laws, incompressible two-fluid flow

@article{M2AN_2005__39_1_37_0, author = {Sever, Michael}, title = {Solutions of a nonhyperbolic pair of balance laws}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {37--58}, publisher = {EDP-Sciences}, volume = {39}, number = {1}, year = {2005}, doi = {10.1051/m2an:2005003}, zbl = {1080.35092}, mrnumber = {2136199}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2005003/} }

TY - JOUR AU - Sever, Michael TI - Solutions of a nonhyperbolic pair of balance laws JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2005 SP - 37 EP - 58 VL - 39 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2005003/ DO - 10.1051/m2an:2005003 LA - en ID - M2AN_2005__39_1_37_0 ER -

Sever, Michael. Solutions of a nonhyperbolic pair of balance laws. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 (2005) no. 1, pp. 37-58. doi : 10.1051/m2an:2005003. http://www.numdam.org/articles/10.1051/m2an:2005003/

[1] Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162 (2002) 327-366. | Zbl

, and ,[2] Hyperbolic Systems of Conservation Laws: the One-dimensional Cauchy Problem. Oxford University Press (2000). | MR | Zbl

,[3] Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28 (1979) 202-212. | Zbl

,[4] 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).

, and ,[5] Theory of Multicomponent Fluids. Springer, New York (1999). | MR | Zbl

and ,[6] An interesting class of quasilinear systems. Dokl. Akad. Nauk SSR 139 (1961) 521-523. | Zbl

,[7] Thermo-fluid dynamic theory of two-phase flow. Eyrolles, Paris (1975). | Zbl

,[8] Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete Contin. Dynam. Systems, Series B 3 (2003) 541-563. | Zbl

, and ,[9] Viscous singular shock structure for a nonhyperbolic two-fluid model. Nonlinearity 17 (2004) 1731-1747. | Zbl

, and ,[10] Parabolic problems which are ill-posed in the zero dissipation limit. Math. Comput. Model. 35 (2002) 1271-1295. | Zbl

and ,[11] Systems of conservation laws of mixed type. J. Diff. Equations 37 (1980) 70-88. | Zbl

,[12] Hyperbolic two-pressure models for two-phase flow. J. Comput. Phys. 53 (1984) 124-151. | Zbl

and ,[13] Computations with singular shocks (2005) (preprint).

and ,[14] A model of discontinuous, incompressible two-phase flow (2005) (preprint).

,[15] Two-phase flow: models and methods. J. Comput. Phys. 56 (1984) 363-409. | Zbl

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