Uncertainty quantification in hierarchical vehicular flow models

Michael Herty; Elisa Iacomini

doi:10.3934/krm.2022006

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April 2022, 15(2): 239-256. doi: 10.3934/krm.2022006

Uncertainty quantification in hierarchical vehicular flow models

Michael Herty and Elisa Iacomini ^,

RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany

^*Corresponding author: Elisa Iacomini

Received July 2021 Revised December 2021 Published April 2022 Early access February 2022

We consider kinetic vehicular traffic flow models of BGK type [24]. Considering different spatial and temporal scales, those models allow to derive a hierarchy of traffic models including a hydrodynamic description. In this paper, the kinetic BGK–model is extended by introducing a parametric stochastic variable to describe possible uncertainty in traffic. The interplay of uncertainty with the given model hierarchy is studied in detail. Theoretical results on consistent formulations of the stochastic differential equations on the hydrodynamic level are given. The effect of the possibly negative diffusion in the stochastic hydrodynamic model is studied and numerical simulations of uncertain traffic situations are presented.

Keywords: Traffic flow, BGK models, stochastic Galerkin, Aw-Rascle-Zhang model, kinetic equations.

Mathematics Subject Classification: Primary: 35R60, 90B20, 35Q20; Secondary: 35Q70.

Citation: Michael Herty, Elisa Iacomini. Uncertainty quantification in hierarchical vehicular flow models. Kinetic and Related Models, 2022, 15 (2) : 239-256. doi: 10.3934/krm.2022006

References:

[1]	A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.
[2]	I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034. doi: 10.1137/050645142.
[3]	M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042.
[4]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.
[5]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.
[6]	R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math., 78 (2018), 2891-2917. doi: 10.1137/17M1152681.
[7]	R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math., 48 (1947), 385-392. doi: 10.2307/1969178.
[8]	J. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531. doi: 10.4208/cicp.oa-2017-0244.
[9]	J. Carrillo and M. Zanella, Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties, Vietnam J. Math., 47 (2019), 931-954. doi: 10.1007/s10013-019-00374-2.
[10]	Q.-Y. Chen, D. Gottlieb and J. S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle, J. Comput. Phys., 204 (2005), 378-398. doi: 10.1016/j.jcp.2004.10.019.
[11]	R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.
[12]	E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413. doi: 10.3934/nhm.2016002.
[13]	B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem and O. P. L. Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26 (2004), 698-719. doi: 10.1137/S1064827503427741.
[14]	B. Després, G. Poëtte and D. Lucor, Uncertainty quantification for systems of conservation laws, J. Comput. Phys., 228 (2009), 2443-2467. doi: 10.1016/j.jcp.2008.12.018.
[15]	M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.
[16]	S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239.
[17]	D. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.
[18]	S. Gerster and M. Herty, Entropies and symmetrization of hyperbolic stochastic Galerkin formulations, Commun. Comput. Phys., 27 (2020), 639-671. doi: 10.4208/cicp.OA-2019-0047.
[19]	S. Gerster, M. Herty and E. Iacomini, Stability analysis of a hyperbolic stochastic galerkin formulation for the aw-rascle-zhang model with relaxation, Math. Biosci. Eng., 18 (2021), 4372-4389. doi: 10.3934/mbe.2021220.
[20]	S. Gerster, M. Herty and A. Sikstel, Hyperbolic stochastic Galerkin formulation for the $p$-system, J. Comput. Phys., 395 (2019), 186-204. doi: 10.1016/j.jcp.2019.05.049.
[21]	D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Commun. Comput. Phys., 3 (2008), 505-518.
[22]	M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311.
[23]	M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.
[24]	M. Herty, G. Puppo, S. Roncoroni and G. Visconti, The BGK approximation of kinetic models for traffic, Kinet. Relat. Models, 13 (2020), 279-307. doi: 10.3934/krm.2020010.
[25]	H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models—a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722. doi: 10.3934/dcds.2018031.
[26]	J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168. doi: 10.1016/j.jcp.2016.03.047.
[27]	S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, SIAM J. Math. Anal., 50 (2018), 1790-1816. doi: 10.1137/17M1123845.
[28]	A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.
[29]	A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114. doi: 10.1007/BF02181481.
[30]	J. Kusch, G. Alldredge and M. Frank, Maximum-principle-satisfying second-order intrusive polynomial moment scheme, SMAI J. Comput. Math., 5 (2019), 23-51. doi: 10.5802/smai-jcm.42.
[31]	O. P. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.
[32]	P. L'Ecuyer and C. Lemieux, Recent advances in randomized quasi-monte carlo methods, Internat. Ser. Oper. Res. Management Sci., 46 (2002), 419-474. doi: 10.1007/0-306-48102-2_20.
[33]	M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.
[34]	O. P. L. Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer Netherlands, 1 ed., 2010. doi: 10.1007/978-90-481-3520-2.
[35]	P. Pettersson, G. Iaccarino and J. Nordström, A stochastic Galerkin method for the Euler equations with Roe variable transformation, J. Comput. Phys., 257 (2014), 481-500. doi: 10.1016/j.jcp.2013.10.011.
[36]	P. Pettersson, G. Iaccarino and J. Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations, Numerical techniques for fluid dynamics problems in the presence of uncertainties. Mathematical Engineering. Springer, Cham, 2015. doi: 10.1007/978-3-319-10714-1.
[37]	R. Pulch and D. Xiu, Generalised polynomial chaos for a class of linear conservation laws, J. Sci. Comput., 51 (2012), 293-312. doi: 10.1007/s10915-011-9511-5.
[38]	G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854. doi: 10.3934/krm.2017033.
[39]	B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772. doi: 10.3934/nhm.2013.8.745.
[40]	R. Shu, J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numer. Math. Theory Methods Appl., 10 (2017), 465-488. doi: 10.4208/nmtma.2017.s12.
[41]	T. J. Sullivan, Introduction to Uncertainty Quantification, Texts in Applied Mathematics, 63. Springer, Cham, 2015. doi: 10.1007/978-3-319-23395-6.
[42]	K. Taimre, Botev, Handbook of Monte Carlo Methods, John Wiley and Sons, 2011.
[43]	A. Tosin and M. Zanella, Boltzmann-type models with uncertain binary interactions, Commun. Math. Sci., 16 (2018), 963-985. doi: 10.4310/CMS.2018.v16.n4.a3.
[44]	A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713. doi: 10.3934/mcrf.2021018.
[45]	N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), 897-936. doi: 10.2307/2371268.
[46]	D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, N.J, 2010.
[47]	D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644. doi: 10.1137/S1064827501387826.
[48]	M. Zanella, Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions, Math. Comput. Simulation, 168 (2020), 28-47. doi: 10.1016/j.matcom.2019.07.012.
[49]	H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.
[50]	Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic preserving method, Multiscale Model. Simul., 15 (2017), 1502-1529. doi: 10.1137/16M1090028.

show all references

References:

[1]	A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.
[2]	I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034. doi: 10.1137/050645142.
[3]	M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042.
[4]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.
[5]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.
[6]	R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math., 78 (2018), 2891-2917. doi: 10.1137/17M1152681.
[7]	R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math., 48 (1947), 385-392. doi: 10.2307/1969178.
[8]	J. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531. doi: 10.4208/cicp.oa-2017-0244.
[9]	J. Carrillo and M. Zanella, Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties, Vietnam J. Math., 47 (2019), 931-954. doi: 10.1007/s10013-019-00374-2.
[10]	Q.-Y. Chen, D. Gottlieb and J. S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle, J. Comput. Phys., 204 (2005), 378-398. doi: 10.1016/j.jcp.2004.10.019.
[11]	R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.
[12]	E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413. doi: 10.3934/nhm.2016002.
[13]	B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem and O. P. L. Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26 (2004), 698-719. doi: 10.1137/S1064827503427741.
[14]	B. Després, G. Poëtte and D. Lucor, Uncertainty quantification for systems of conservation laws, J. Comput. Phys., 228 (2009), 2443-2467. doi: 10.1016/j.jcp.2008.12.018.
[15]	M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.
[16]	S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239.
[17]	D. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.
[18]	S. Gerster and M. Herty, Entropies and symmetrization of hyperbolic stochastic Galerkin formulations, Commun. Comput. Phys., 27 (2020), 639-671. doi: 10.4208/cicp.OA-2019-0047.
[19]	S. Gerster, M. Herty and E. Iacomini, Stability analysis of a hyperbolic stochastic galerkin formulation for the aw-rascle-zhang model with relaxation, Math. Biosci. Eng., 18 (2021), 4372-4389. doi: 10.3934/mbe.2021220.
[20]	S. Gerster, M. Herty and A. Sikstel, Hyperbolic stochastic Galerkin formulation for the $p$-system, J. Comput. Phys., 395 (2019), 186-204. doi: 10.1016/j.jcp.2019.05.049.
[21]	D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Commun. Comput. Phys., 3 (2008), 505-518.
[22]	M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311.
[23]	M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.
[24]	M. Herty, G. Puppo, S. Roncoroni and G. Visconti, The BGK approximation of kinetic models for traffic, Kinet. Relat. Models, 13 (2020), 279-307. doi: 10.3934/krm.2020010.
[25]	H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models—a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722. doi: 10.3934/dcds.2018031.
[26]	J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168. doi: 10.1016/j.jcp.2016.03.047.
[27]	S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, SIAM J. Math. Anal., 50 (2018), 1790-1816. doi: 10.1137/17M1123845.
[28]	A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.
[29]	A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114. doi: 10.1007/BF02181481.
[30]	J. Kusch, G. Alldredge and M. Frank, Maximum-principle-satisfying second-order intrusive polynomial moment scheme, SMAI J. Comput. Math., 5 (2019), 23-51. doi: 10.5802/smai-jcm.42.
[31]	O. P. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.
[32]	P. L'Ecuyer and C. Lemieux, Recent advances in randomized quasi-monte carlo methods, Internat. Ser. Oper. Res. Management Sci., 46 (2002), 419-474. doi: 10.1007/0-306-48102-2_20.
[33]	M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.
[34]	O. P. L. Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer Netherlands, 1 ed., 2010. doi: 10.1007/978-90-481-3520-2.
[35]	P. Pettersson, G. Iaccarino and J. Nordström, A stochastic Galerkin method for the Euler equations with Roe variable transformation, J. Comput. Phys., 257 (2014), 481-500. doi: 10.1016/j.jcp.2013.10.011.
[36]	P. Pettersson, G. Iaccarino and J. Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations, Numerical techniques for fluid dynamics problems in the presence of uncertainties. Mathematical Engineering. Springer, Cham, 2015. doi: 10.1007/978-3-319-10714-1.
[37]	R. Pulch and D. Xiu, Generalised polynomial chaos for a class of linear conservation laws, J. Sci. Comput., 51 (2012), 293-312. doi: 10.1007/s10915-011-9511-5.
[38]	G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854. doi: 10.3934/krm.2017033.
[39]	B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772. doi: 10.3934/nhm.2013.8.745.
[40]	R. Shu, J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numer. Math. Theory Methods Appl., 10 (2017), 465-488. doi: 10.4208/nmtma.2017.s12.
[41]	T. J. Sullivan, Introduction to Uncertainty Quantification, Texts in Applied Mathematics, 63. Springer, Cham, 2015. doi: 10.1007/978-3-319-23395-6.
[42]	K. Taimre, Botev, Handbook of Monte Carlo Methods, John Wiley and Sons, 2011.
[43]	A. Tosin and M. Zanella, Boltzmann-type models with uncertain binary interactions, Commun. Math. Sci., 16 (2018), 963-985. doi: 10.4310/CMS.2018.v16.n4.a3.
[44]	A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713. doi: 10.3934/mcrf.2021018.
[45]	N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), 897-936. doi: 10.2307/2371268.
[46]	D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, N.J, 2010.
[47]	D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644. doi: 10.1137/S1064827501387826.
[48]	M. Zanella, Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions, Math. Comput. Simulation, 168 (2020), 28-47. doi: 10.1016/j.matcom.2019.07.012.
[49]	H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.
[50]	Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic preserving method, Multiscale Model. Simul., 15 (2017), 1502-1529. doi: 10.1137/16M1090028.

Figure 1. Outline of the model hierarchy. The left two columns indicate the kinetic and fluid description of traffic flow as presented in [24]. The third column refers to the diffusion coefficient $ \mu(\rho) $ to classify traffic instabilities. The green hierarchy is deterministic while the blue includes a parametric uncertainty $ \xi. $ The indicated links are established in this paper

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Figure 2. Probability (53) for a Maxwellian $ M_f $ with different number of discrete velocities: $ n_v = 3 $, $ n_v = 10 $. On the x-axis $ \rho_0 $ is shown, see (61)

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Figure 3. Probability (53) for different velocities samples: $ n_v = 3 $ (blue line), $ n_v = 10 $(red line) for different values of $ \rho_0 = 0.4 $(left) $ \rho_0 = 0.6 $(right) when the standard deviation $ \sigma $ ranges from zero to $ 0.2 $

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Figure 4. Probability (53) for different hesitation functions: $ h(\rho) = \rho $(blue line) and $ h(\rho) = \rho^3 $(red line)

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Figure 5. Mean and variance of the density at $ t = T_f $ for different

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Figure 6. Probability (53) at time $ t = 0 $ (black line), $ t = \frac{{T_f}}{2} $ (green line) $ t = T $ (blue line) for $ K = 4 $ (top-left), $ K = 8 $ (top-right), $ K = 32 $ (bottom-left), $ K = 64 $ (bottom-right)

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Figure 7. Probability of negative diffusion coefficient in a rarefaction case at different time: $ t = 0 $, $ t = \frac{T_f}{2} $, $ t = T_f $, $ K = 64 $, and comparison with the confident region of the density at $ t = T_f $

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Figure 8. Density profile and probability of negative diffusion coefficient in a shock case at $ t = T_f $, $ K = 64 $

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2021 Impact Factor: 1.398

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2021 Impact Factor: 1.398

American Institute of Mathematical Sciences

Uncertainty quantification in hierarchical vehicular flow models

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American Institute of Mathematical Sciences

Uncertainty quantification in hierarchical vehicular flow models

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