Abstract
Arguments are advanced that physically meaningful nonunique solutions to Riemann problems can occur. The implications of this point of view for both theory and computation are developed, as part of a review of recent progress concerning the interaction of nonlinear waves and the front tracking method for computation.
Supported in part by the National Science Foundation, grant DMS - 8619856
Supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U. S. Department of Energy, under contract DE-AC02-76ER03077
Supported in part by the Army Research Office, grant DAAG29-85-0188
Supported in part by the Air Force Office Office of Scientific Research AFSOR-88-0025.
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References
M. Brio, “Admissibility Conditions for Weak Conditions of Non-strictly Hyperbolic Systems”, paper in this volume.
B. Bukiet and J. Jones, “The Competition Between Curvature and Chemistry in a Spherically Expanding Detonation”, Appl. Phys. Leteers, in press.
C. Conley and J. Smoller, “Viscosity Matrices for Two-Dimensional Nonlinear Hyperbolic Systems”, Comm. Pure Appl. Math., vol. 23, pp. 867–884, 1970.
P. Daripa, J. Glimm, B. Lindquist, and O. McBryan, “Polymer floods: a case study of nonlinear wave analysis and of instability control in tertiary oil recovery”, SIAM J Appl. Math, vol. 48, pp. 353–373, 1988.
H. Freistühler, “A Standard Model of Generic Rotational Degeneracy”, paper in this volume. Conference on Hyperbolic Problems, To appear.
C. Gardner, J. Glimm, O. McBryan, R. Menikoff, D. H. Sharp, and Q. Zhang, “The Dynamics of Bubble Growth for Rayleigh-Taylor Unstable Interfaces,” Phys. of Fluids, vol. 31, pp. 447–465, 1988.
J. Glimm, B. Lindquist, O. McBryan, and L. Padmanhaban, “A Front Tracking Reservoir Simulator: 5-spot Validation Studies and the Water Coning Problem,” Frontiers in Applied Mathematics, vol. 1, SIAM, Philadelphia, 1983.
J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp, and S. Yaniv, “Front Tracking and Two Dimensional Riemann Problems,” Advances in Appl. Math., vol. 6, pp. 259–290, 1985.
J. Glimm, O. McBryan, D. Sharp, and R. Menikoff, “Front Tracking Applied to Rayleigh Taylor Instability,” SIAM J. Sci. Stat. Comput., vol. 7, pp. 230–251, 1986.
J. Glimm, J. Grove, and X.L. Li, “Three Remarks on the Front Tracking Method,” in Proceedings of Taormina Conference, Oct 1987.
J. Glimm, B. Lindquist, O. McBryan, and G. Tryggvason, “Sharp and Diffuse Fronts in Oil Reservoirs: Front Tracking and Capillarity,” in Mathematical and Computational Methods in Seismic Exploration and Reservoir Modeling, ed. William E. Fitzgibbon, pp. 68–84, SIAM, Philadelphia, 1987.
J. Glimm, “The Interactions of Nonlinear Hyperbolic Waves,” Comm. Pure Appl. Math., vol. 41, pp. 569–590, 1988.
J. Glimm and X.L. Li, “On the Validation of the Sharp-Wheeler Bubble Merger Model from Experimental and Computational Data,” Phys. of Fluids, vol. 31, pp. 2077–2085, 1988.
Maria Elasir Gomes, “Singular Riemann Problem for a Fourth Order Model for Multiphase Flow,” Thesis (Portuguese) Departamento de Matematica, Pontificia Universidade Catolica do Rio Janeiro, 1987.
Eli Isaacson, D. Marchesin, B. Plohr, and B. Temple, “The Classification of Solutions of Quadratic Riemann Problems I,” MRC Report, 1985.
Eli Isaacson and B. Temple, “The Classification of Solutions of Quadratic Riemann Problems H, HI,” SIAM J Appl. Math., To appear.
Eli Isaacson and B. Temple, The Structure of Asymptotic States in a Singular System of Conservation Laws, To Appear.
Eli Isaacson, D. Marchesin, and B. Plohr, Viscous Profiles for Transitional Shock Waves, To Appear.
B. Lindquist, “The Scalar Riemann Problem in Two Spatial Dimensions: Piecewise Smoothness of Solutions and its Breakdown,” SIAM J Anal, vol. 17, pp. 1178–1197, 1986.
J. Marsden and T. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, 1983.
R. Menikoff and B. Plohr, “Riemann Problem for Fluid Flow of Real Materials,” Rev. Mod. Phys., To Appear.
B. Plohr and D. H. Sharp, “A Conservative Eulerian Formulation for the Equation of Elastic Flow,” Adv. Appl. Math., To appear.
K. I. Read, “Experimental Investigation of Turbulent Mixing by Rayleigh-Taylor Instability,” Physica 12D, pp. 45–48, 1984.
D. H. Sharp and J. A. Wheeler, “Late Stage of Rayleigh-Taylor Instability,” Institute for Defense Analyses, 1961.
D. H. Sharp, “Overview of Rayleigh-Taylor Instability,” Physica 12D, pp. 3–17, 1984.
S. Stewart and J. B. Bdzil, “The Shock Dynamics of Stable Multidimensional Detonation,” J. Fluid Mech., To appear.
D. Wagner, “The Riemann Problem in Two Space Dimensions for a Single Conservation Law,” Math. Ann., vol. 14, pp. 534–559, 1983.
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© 1989 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Glimm, J. (1989). Nonuniqueness of Solutions for Riemann Problems. In: Ballmann, J., Jeltsch, R. (eds) Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications. Notes on Numerical Fluid Mechanics, vol 24. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-87869-4_17
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DOI: https://doi.org/10.1007/978-3-322-87869-4_17
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