Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation

Stephan Gerster; Michael Herty; Elisa Iacomini; Stephan Gerster; Michael Herty; Elisa Iacomini

doi:10.3934/mbe.2021220

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 4372-4389. doi: 10.3934/mbe.2021220

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Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation

RWTH Aachen University, Institute for Geometry and Applied Mathematics, Aachen, Germany

Received: 12 February 2021 Accepted: 06 May 2021 Published: 20 May 2021

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.
- traffic flow,
- uncertainty quantification,
- stability analysis,
- Aw-Rascle-Zhang model,
- stochastic Galerkin,
- Chapman-Enskog expansion
Citation: Stephan Gerster, Michael Herty, Elisa Iacomini. Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4372-4389. doi: 10.3934/mbe.2021220

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Abstract

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.

References

[1]	W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics–I, Bull. Math. Biol., 53 (1991), 33–55.
[2]	J. Noble, Geographic and temporal development of plagues, Nature, 250 (1974), 726–729. doi: 10.1038/250726a0
[3]	G. Puppo, M. Semplice, A. Tosin, G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), 643–669. doi: 10.4310/CMS.2016.v14.n3.a3
[4]	M. Herty, A. Tosin, G. Visconti, M. Zanella, Reconstruction of traffic speed distributions from kinetic models with uncertainties, SEMA SIMAI Springer Series, 2020.
[5]	M. Herty, A. Tosin, G. Visconti, M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, SIAM J. Appl. Math., 78 (2018), 2737–2762. doi: 10.1137/17M1155909
[6]	A. Tosin, M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control. Relat. Fields, 78 (2021), 2737–2762.
[7]	O. P. L. Maître, O. M. Knio, Spectral Methods for uncertainty quantification, 1st edition, Springer Netherlands, 2010.
[8]	R. H. Cameron, W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. Math., 48 (1947), 385–392. doi: 10.2307/1969178
[9]	R. G. Ghanem, P. D. Spanos, Stochastic finite elements: A Spectral Approach, 1st edition, Springer, New York, 1991.
[10]	D. Gottlieb, J. S. Hesthaven, Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128 (2001), 83–131. doi: 10.1016/S0377-0427(00)00510-0
[11]	N. Wiener, The homogeneous chaos, Am. J. Math., 60 (1938), 897–936.
[12]	D. Xiu, G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619–644.
[13]	B. Després, G. Poëtte, D. Lucor, Uncertainty quantification for systems of conservation laws, J. Comput. Phys., 228 (2009), 2443–2467. doi: 10.1016/j.jcp.2008.12.018
[14]	S. Jin, R. Shu, A study of hyperbolicity of kinetic stochastic Galerkin system for the isentropic Euler equations with uncertainty, Chinese Ann. Math. Ser. B, 40 (2019), 765–780. doi: 10.1007/s11401-019-0159-z
[15]	B. Després, G. Poëtte, D. Lucor, Robust uncertainty propagation in systems of conservation laws with the entropy closure method, vol. 92 of Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, Springer, Cham, 2013.
[16]	P. Pettersson, G. Iaccarino, 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
[17]	S. Jin, D. Xiu, X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198–1218. doi: 10.1007/s10915-015-0124-2
[18]	J. Kusch, G. W. Alldredge, M. Frank, Maximum-principle-satisfying second-order intrusive polynomial moment scheme, J. Comput. Math., 5 (2019), 23–51.
[19]	J. Kusch, R. G. McClarren, M. Frank, Filtered stochastic Galerkin methods for hyperbolic equations, J. Comput. Phys., 403 (2020), 109073. doi: 10.1016/j.jcp.2019.109073
[20]	J. Kusch, J. Wolters, M. Frank, Intrusive acceleration strategies for uncertainty quantification for hyperbolic systems of conservation laws, J. Comput. Phys., 419 (2020), 109698. doi: 10.1016/j.jcp.2020.109698
[21]	D. Dai, Y. Epshteyn, A. Narayan, Hyperbolicity-preserving and well-balanced stochastic Galerkin method for shallow water equations, SIAM J. Appl. Math., 43 (2021), A929–A952.
[22]	O. P. L. Maître, O. M. Knio, H. N. Najm, R. G. Ghanem, Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys., 197 (2004), 28–57. doi: 10.1016/j.jcp.2003.11.033
[23]	R. Abgrall, P. Congedo, G. Geraci, G. Iaccarino, An adaptive multiresolution semi-intrusive scheme for UQ in compressible fluid problems, Int. J. Numer. Methods Fluids, 78 (2015), 595–637. doi: 10.1002/fld.4030
[24]	I. Kröker, W. Nowak, C. Rohde, A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems, Comput. Geosci., 19 (2015), 269–284. doi: 10.1007/s10596-014-9464-5
[25]	J. Tryoen, O. P. L. Maître, O. M. Knio, A. Ern, Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws, SIAM J. Sci. Comput., 34 (2012), 2459–2481. doi: 10.1137/120863927
[26]	A. Aw, M. Rascle, Resurrection of "second order" models of traffic flow?, SIAM J. Appl. Math., 60 (2000), 916–938. doi: 10.1137/S0036139997332099
[27]	H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-meth., 36 (2002), 275–290. doi: 10.1016/S0191-2615(00)00050-3
[28]	J. Greenberg, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175–1185.
[29]	M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales, B. Seibold, Self-sustained nonlinear waves in traffic flow, Physical Review E, 79 (2009), 056113. doi: 10.1103/PhysRevE.79.056113
[30]	M. Herty, G. Puppo, G. Visconti, From kinetic to macroscopic models and back, SEMA SIMAI Springer Series, 1–14.
[31]	G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience, 431–484.
[32]	J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2002), 729–745. doi: 10.1137/S0036139900378657
[33]	M. J. Lighthill, G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Math. Phys. Eng. Sci., 229 (1955), 317–345.
[34]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42–51.
[35]	G. Q. Chen, C. D. Levermore, T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47 (1994), 787–830. doi: 10.1002/cpa.3160470602
[36]	S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48 (1995), 235–276. doi: 10.1002/cpa.3160480303
[37]	B. Seibold, M. R. Flynn, A. R. Kasimov, 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
[38]	M. Herty, G. Puppo, S. Roncoroni, G. Visconti, The BGK approximation of kinetic models for traffic, Kinet. Relat. Models, 13 (2020), 279. doi: 10.3934/krm.2020010
[39]	S. Fan, M. Herty, B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2013), 239–268.
[40]	O. G. Ernst, A. Mugler, H. J. Starkloff, E. Ullmann, On the convergence of generalized polynomial chaos expansions, ESAIM: M2AN, 46 (2012), 317–339. doi: 10.1051/m2an/2011045
[41]	D. Funaro, Polynomial approximation of differential equations, vol. 8, Springer Science & Business Media, 2008.
[42]	B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem, 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
[43]	T. J. Sullivan, Introduction to uncertainty quantification, 1st edition, Texts in Applied Mathematics, Springer, Switzerland, 2015.
[44]	B. Sonday, R. Berry, H. Najm, B. Debusschere, Eigenvalues of the Jacobian of a Galerkin-projected uncertain ODE system, J. Sci. Comput., 33 (2011), 1212–1233.
[45]	K. Wu, H. Tang, D. Xiu, A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty, J. Comput. Phys., 345 (2017), 224–244. doi: 10.1016/j.jcp.2017.05.027
[46]	S. Gerster, M. Herty, 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
[47]	S. Gerster, M. Herty, Entropies and symmetrization of hyperbolic stochastic Galerkin formulations, Commun. Comput. Phys., 27 (2020), 639–671. doi: 10.4208/cicp.OA-2019-0047
[48]	S. Gerster, Stabilization and uncertainty quantification for systems of hyperbolic balance laws, Dissertation, RWTH Aachen University, Aachen, 2020.
[49]	A. Bressan, Hyperbolic systems of conservation laws: The one dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 2005.
[50]	B. Gustafsson, H.-O. Kreiss, J. Oliger, Time-dependent problems and difference methods, 2nd edition, Wiley, 2013.
[51]	B. D. Greenshields, A study of traffic capacity, in Proceedings of the highway research board, vol. 14, 1935,448–477.
[52]	F. Siebel, W. Mauser, On the fundamental diagram of traffic flow, SIAM J. Appl. Math., 66 (2005), 1150–1162.
[53]	A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69 (1910), 331–371. doi: 10.1007/BF01456326
[54]	P. Pettersson, G. Iaccarino, J. Nordström, Polynomial chaos methods for hyperbolic partial differential equations, Springer International Publishing, Switzerland, 2015.
[55]	R. J. Leveque, Numerical Methods for Conservation Laws, 2nd edition, Lectures in Mathematics. ETH Zürich, Birkhäuser Basel, 1992.
[56]	L. Pareschi, G. Russo, Implicit–explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129–155.
[57]	S. Pieraccini, G. Puppo, Implicit–explicit Schemes for BGK kinetic equations, J. Sci. Comput., 32 (2007), 1–28. doi: 10.1007/s10915-006-9116-6
[58]	P. Pettersson, G. Iaccarino, J. Nordström, Numerical analysis of the Burgers' equation in the presence of uncertainty, J. Comput. Phys., 228 (2009), 8394–8412. doi: 10.1016/j.jcp.2009.08.012

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