The Riemann problem for a spray model

References

• Baer MR, Nunziato JW. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int J Multiphase Flows. 1986;12:861–889.
   Web of Science ®Google Scholar
• Bzil JB, Menikoff R, Son SF, et al. Two-phase modeling of a deflagration-to-detonation transition in granular materials: a critical examination of modelling issues. Phys Fluids. 1999;11:378–402.
   Web of Science ®Google Scholar
• Dal Maso G, LeFloch PG, Murat F. Definition and weak stability of nonconservative products. J Math Pures Appl. 1995;74(9):483–548.
   Google Scholar
• LeFloch PG, Tzavaras AE. Representation of weak limits and definition of nonconservative products. SIAM J Math Anal. 1999;30:1309–1342.
   Web of Science ®Google Scholar
• Isaacson E, Temple B. Nonlinear resonance in systems of conservation laws. SIAM J Appl Math. 1992;52:1260–1278.
   Web of Science ®Google Scholar
• Isaacson E, Temple B. Convergence of the 2×2 Godunov method for a general resonant nonlinear balance law. SIAM J Appl Math. 1995;55:625–640.
   Web of Science ®Google Scholar
• LeFloch PG. Shock waves for nonlinear hyperbolic systems in nonconservative form. Minneapolis: Institute for Mathematics and its Application; Preprint 593, 1989.
   Google Scholar
• Marchesin D, Paes-Leme PJ. A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III. Comput Math Appl Part A. 1986;12:433–455.
   Google Scholar
• Ripa P. Conservation laws for primitive equations models with inhomogeneous layers. Geophys Astrophys Fluid Dyn. 1993;70:85–111.
   Web of Science ®Google Scholar
• Ripa P. On improving a one-layer ocean model with thermodynamics. J Fluid Mech. 1995;303:169–201.
   Web of Science ®Google Scholar
• Andrianov N, Warnecke G. On the solution to the Riemann problem for the compressible duct flow. SIAM J Appl Math. 2004;64:878–901.
   Web of Science ®Google Scholar
• LeFloch PG, Thanh MD. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun Math Sci. 2003;1:763–797.
   Google Scholar
• Thanh MD. The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J Appl Math. 2009;69:1501–1519.
   Web of Science ®Google Scholar
• Andrianov N, Warnecke G. The Riemann problem for the Baer-Nunziato two-phase flow model. J Comput Phys. 2004;195:434–464.
   Web of Science ®Google Scholar
• Thanh MD. A phase decomposition approach and the Riemann problem for a model of two-phase flows. J Math Anal Appl. 2014;418:569–594.
   Web of Science ®Google Scholar
• Keyfitz BL, Sander R, Sever M. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete Contin Dyn Syst Ser B. 2003;3:541–563.
   Web of Science ®Google Scholar
• Rosatti G, Begnudelli L. The Riemann problem for the one-dimensional, free-surface shallow water equations with a bed step: theoretical analysis and numerical simulations. J Comput Phys. 2010;229:760–787.
   Web of Science ®Google Scholar
• Thanh MD. The Riemann problem for the shallow water equations with horizontal temperature gradients. Appl Math Comput. 2018;325:159–178.
   Web of Science ®Google Scholar
• Goatin P, LeFloch PG. The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann Inst H Poincaré Anal Non Linéaire. 2004;21:881–902.
   Web of Science ®Google Scholar
• Ambroso A, Chalons C, Raviart P-A. A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput Fluids. 2012;54:67–91.
   Web of Science ®Google Scholar
• Cuong DH, Thanh MD. A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section. Appl Math Comput. 2015;256:602–629.
   Web of Science ®Google Scholar
• Schwendeman DW, Wahle CW, Kapila AK. The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J Comput Phys. 2006;212:490–526.
   Web of Science ®Google Scholar
• Shao Z. The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation. Z Angew Math Phys. 2018;69:44.
   Web of Science ®Google Scholar
• Shao Z. Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations. Z Angew Math Phys. 2016;67:66.
   Web of Science ®Google Scholar