Abstract
The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data.
Similar content being viewed by others
References
Armbruster, D., Degond, P., Ringhofer, C.: A model for the dynamics of large queuing metworks and supply chains. SIAM J. Appl. Math. 66, 896–920 (2006)
Herty, M., Klar, A., Piccoli, B.: Existence of solutions for supply chain models based on partial differential equations. SIAM J. Math. Anal. 39, 160–173 (2007)
Forestier-Coste, L., Gottlich, S., Herty, M.: Data-fitted second-order macroscopic production models. SIAM J. Appl. Math. 75, 999–1014 (2015)
Sun, M.: Singular solutions to the Riemann problem for a macroscopic production model. Z. Angew. Math. Mech. 97, 916–931 (2017)
Armbruster, D., Marthaler, D., Ringhofer, C.: Kinetic and fluid model heirarchies for supply chains. Multiscale Model. Simul. 2, 43–61 (2003)
Armbruster, D., Wienke, M.: Kinetic models and intrinsic timescales: simulation comparison for a 2nd order queueing model. Kinet. Relat. Models 12, 177–193 (2019)
Brenier, Y.: Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J. Math. Fluid Mech. 7, S326–S331 (2005)
Lai, G., Sheng, W., Zheng, Y.: Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions. Discrete Contin. Dyn. Syst. 31, 489–523 (2011)
Guo, L., Zhang, Y., Yin, G.: Interactions of delta shock waves for the Chaplygin gas equations with split delta functions. J. Math. Anal. Appl. 410, 190–201 (2014)
Shen, C.: The Riemann problem for the Chaplygin gas equations with a source term. Z. Angew. Math. Mech. 96, 681–695 (2016)
Shao, Z.: Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations, Z. Angew. Math. Phys., 67 (2016), Article ID 66
Pan, L., Han, X.: The Aw-Rascle traffic model with Chaplygin pressure. J. Math. Anal. Appl. 401, 379–387 (2013)
Zeidan, D., Romenski, E., Slaouti, A., Toro, E.F.: Numerical study of wave propagation in compressible two-phase flow. Int. J. Numer. Meth. Fluids 54, 393–417 (2007)
Goncalves, E., Hoarau, Y., Zeidan, D.: Simulation of shock-induced bubble collapse using a four-equation model. Shock Waves 29, 221–234 (2019)
Zeidan, D., Bähr, P., Farber, P., Gräbel, J., Ueberholz, P.: Numerical investigation of a mixture two-phase flow model in two-dimensional space. Comput. Fluids 181, 90–106 (2019)
Toro, E.F.: Riemann solves and numerical methods for fluid dynamics: a practical introduction. Springer Science and Business Media, Berlin (2013)
Zeidan, D., Zhang, L.T., Goncalves, E.: High-resolution simulations for aerogel using two-phase flow equations and Godunov methods. Int. J. Appl. Mech. 12, 2050049 (2020)
Zeidan, D., Touma, R.: On the computations of gas-solid mixture two-phase flow. Adv. Appl. Math. Mech. 6, 49–74 (2014)
Temple, B.: Systems of conservation laws with invariant submanifolds. Trans. Am. Math. Soc. 280, 781–795 (1983)
Shen, C., Sun, M.: A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system. Int. J. Non-linear Mech. 105, 105–122 (2018)
R. De la cruz, M. Santos, : Delta shock waves for a system of Keyfitz-Kranzer type. Z. Angew. Math. Mech. 99, e201700251 (2019)
Sheng, W., Zhang, T.: (1999) The Riemann problem for the transportation equations in gas dynamics. Mem. Amer. Math. Soc. 137(N654), AMS: Providence
Chen, G.Q., Liu, H.: Formation of \(\delta \)-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34, 925–938 (2003)
Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of \(\delta \)-shock waves in conservation law systems. J. Differ. Equ. 211, 333–381 (2005)
Nedeljkov, M.: Shadow waves: entropies and interactions for delta and singular shocks. Arch. Ration. Mech. Anal. 197, 489–537 (2010)
Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252, 5951–5993 (2012)
Shen, C.: The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows. Math. Meth. Appl. Sci. 43, 3673–3688 (2020)
Sun, M.: Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state. Nonlinear Anal. RWA 53, 103068 (2020)
Nedeljkov, M., Oberguggenberger, M.: Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J. Math. Anal. Appl. 344, 1143–1157 (2008)
Li, S., Shen, C.: Construction of global Riemann solutions with delta-type initial data for a thin film model with a perfectly soluble anti-surfactant solution. Int. J. Non linear Mech. 120, 103392 (2020)
Li, S., Shen, C.: Measure-valued solutions to a non-strictly hyperbolic system with delta-type Riemann initial data. International Journal of Nonlinear Sciences and Numerical Simulation, https://doi.org/10.1515/ijnsns-2019-0069, in press
Lai, G., Sheng, W.: Elementary wave interactions to the compressible Euler equations for Chaplygin gas in two dimensions. SIAM J. Appl. Math. 76, 2218–2242 (2016)
Raja Sekhar, T., Sharma, V.D.: Riemann problem and elementary wave interactions in isentropic magnetogasdynamics. Nonlinear Anal. RWA 11, 619–636 (2010)
Sun, M., Xin, J.: On the delta shock wave interactions for the isentropic Chaplygin gas system consisting of three scalar equations. Filomat 33, 5355–5373 (2019)
Sen, A., Sekhar, T.R., Sharma, V.D.: Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws. Q. Appl. Math. 75, 539–554 (2017)
Shen, C.: Delta shock wave solution for a symmetric Keyfitz-Kranzer system. Appl. Math. Lett. 77, 35–43 (2018)
Sun, M.: The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type. Math. Meth. Appl. Sci. 43, 682–697 (2020)
Guo, L., Zhang, Y., Yin, G.: Interactions of delta shock waves for the relativistic Chaplygin gas equations with split delta functions. Math. Meth. Appl. Sci. 38, 2132–2148 (2015)
Castaneda, P.: Embedded delta shocks. Heliyon 6, e04152 (2020)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Norhashidah Hj. Mohd. Ali.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partially supported by Shandong Provincial Natural Science Foundation (ZR2019MA058).
Rights and permissions
About this article
Cite this article
Wang, P., Shen, C. The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas. Bull. Malays. Math. Sci. Soc. 44, 1195–1214 (2021). https://doi.org/10.1007/s40840-020-01003-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-01003-9