Construction of boundary conditions for hyperbolic relaxation approximations II: Jin-Xin relaxation model
Authors:
Xiaxia Cao and Wen-An Yong
Journal:
Quart. Appl. Math. 80 (2022), 787-816
MSC (2020):
Primary 35L50
DOI:
https://doi.org/10.1090/qam/1627
Published electronically:
May 25, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This is our second work in the series about constructing boundary conditions for hyperbolic relaxation approximations. The present work is concerned with the one-dimensional linearized Jin-Xin relaxation model, a convenient approximation of hyperbolic conservation laws, with non-characteristic boundaries. Assume that proper boundary conditions are given for the conservation laws. We construct boundary conditions for the relaxation model with the expectation that the resultant initial-boundary-value problems are approximations to the given conservation laws with the boundary conditions. The constructed boundary conditions are highly non-unique. Their satisfaction of the generalized Kreiss condition is analyzed. The compatibility with initial data is studied. Furthermore, by resorting to a formal asymptotic expansion, we prove the effectiveness of the approximations.
References
- Denise Aregba-Driollet and Roberto Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973–2004. MR 1766856, DOI 10.1137/S0036142998343075
- A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000), no. 3, 916–938. MR 1750085, DOI 10.1137/S0036139997332099
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
- R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math. 78 (2018), no. 5, 2891–2917. MR 3867627, DOI 10.1137/17M1152681
- F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. MR 1705583, DOI 10.1023/A:1004525427365
- Zhenning Cai, Yuwei Fan, and Ruo Li, A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math. 75 (2015), no. 5, 2001–2023. MR 3394370, DOI 10.1137/14100110X
- Zhenning Cai and Manuel Torrilhon, Numerical simulation of microflows using moment methods with linearized collision operator, J. Sci. Comput. 74 (2018), no. 1, 336–374. MR 3742882, DOI 10.1007/s10915-017-0442-7
- D. Chakraborty and J. E. Sader, Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales, Phys. Fluids 27 (2015) (5), 52002.
- Alexandre Ern and Vincent Giovangigli, Multicomponent transport algorithms, Lecture Notes in Physics. New Series m: Monographs, vol. 24, Springer-Verlag, Berlin, 1994. MR 1321142
- Renée Gatignol, Théorie cinétique des gaz à répartition discrète de vitesses, Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin-New York, 1975 (French). MR 0416379
- Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger, Time dependent problems and difference methods, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1377057
- Michael Herty and Wen-An Yong, Feedback boundary control of linear hyperbolic systems with relaxation, Automatica J. IFAC 69 (2016), 12–17. MR 3500083, DOI 10.1016/j.automatica.2016.02.016
- Robert L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev. 28 (1986), no. 2, 177–217. MR 839822, DOI 10.1137/1028050
- Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, DOI 10.1002/cpa.3160480303
- C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021–1065. MR 1392419, DOI 10.1007/BF02179552
- Hailiang Liu and Wen-An Yong, Time-asymptotic stability of boundary-layers for a hyperbolic relaxation system, Comm. Partial Differential Equations 26 (2001), no. 7-8, 1323–1343. MR 1855280, DOI 10.1081/PDE-100106135
- Andrew Majda and Stanley Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), no. 5, 607–675. MR 410107, DOI 10.1002/cpa.3160280504
- Guy Métivier, Small viscosity and boundary layer methods, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2004. Theory, stability analysis, and applications. MR 2151414, DOI 10.1007/978-0-8176-8214-9
- Luc Mieussens, Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries, J. Comput. Phys. 162 (2000), no. 2, 429–466. MR 1774264, DOI 10.1006/jcph.2000.6548
- Shinya Nishibata, The initial-boundary value problems for hyperbolic conservation laws with relaxation, J. Differential Equations 130 (1996), no. 1, 100–126. MR 1409025, DOI 10.1006/jdeq.1996.0134
- W. Vincenti and C. Jr. Kruger, Introduction to physical gas dynamic, Krieger, Malabar, 1986.
- Wei-Cheng Wang and Zhouping Xin, Asymptotic limit of initial-boundary value problems for conservation laws with relaxational extensions, Comm. Pure Appl. Math. 51 (1998), no. 5, 505–535. MR 1604274, DOI 10.1002/(SICI)1097-0312(199805)51:5<505::AID-CPA3>3.0.CO;2-C
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- Zhouping Xin and Wen-Qing Xu, Stiff well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane, J. Differential Equations 167 (2000), no. 2, 388–437. MR 1793199, DOI 10.1006/jdeq.2000.3806
- Wen-Qing Xu, Boundary conditions and boundary layers for a multi-dimensional relaxation model, J. Differential Equations 197 (2004), no. 1, 85–117. MR 2030150, DOI 10.1016/j.jde.2003.08.007
- Wen-An Yong, Boundary conditions for hyperbolic systems with stiff source terms, Indiana Univ. Math. J. 48 (1999), no. 1, 115–137. MR 1722195, DOI 10.1512/iumj.1999.48.1611
- Wen-An Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations 155 (1999), no. 1, 89–132. MR 1693210, DOI 10.1006/jdeq.1998.3584
- Wen-An Yong, Newtonian limit of Maxwell fluid flows, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 913–922. MR 3269638, DOI 10.1007/s00205-014-0769-2
- Yizhou Zhou and Wen-An Yong, Construction of boundary conditions for hyperbolic relaxation approximations I: The linearized Suliciu model, Math. Models Methods Appl. Sci. 30 (2020), no. 7, 1407–1439. MR 4122254, DOI 10.1142/S0218202520500268
- Yizhou Zhou and Wen-An Yong, Boundary conditions for hyperbolic relaxation systems with characteristic boundaries of type I, J. Differential Equations 281 (2021), 289–332. MR 4213673, DOI 10.1016/j.jde.2021.02.008
- Yizhou Zhou and Wen-An Yong, Boundary conditions for hyperbolic relaxation systems with characteristic boundaries of type II, J. Differential Equations 310 (2022), 198–234. MR 4355918, DOI 10.1016/j.jde.2021.11.020
References
- Denise Aregba-Driollet and Roberto Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973–2004. MR 1766856, DOI 10.1137/S0036142998343075
- A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000), no. 3, 916–938. MR 1750085, DOI 10.1137/S0036139997332099
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
- R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math. 78 (2018), no. 5, 2891–2917. MR 3867627, DOI 10.1137/17M1152681
- F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. MR 1705583, DOI 10.1023/A:1004525427365
- Zhenning Cai, Yuwei Fan, and Ruo Li, A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math. 75 (2015), no. 5, 2001–2023. MR 3394370, DOI 10.1137/14100110X
- Zhenning Cai and Manuel Torrilhon, Numerical simulation of microflows using moment methods with linearized collision operator, J. Sci. Comput. 74 (2018), no. 1, 336–374. MR 3742882, DOI 10.1007/s10915-017-0442-7
- D. Chakraborty and J. E. Sader, Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales, Phys. Fluids 27 (2015) (5), 52002.
- Alexandre Ern and Vincent Giovangigli, Multicomponent transport algorithms, Lecture Notes in Physics. New Series m: Monographs, vol. 24, Springer-Verlag, Berlin, 1994. MR 1321142
- Renée Gatignol, Théorie cinétique des gaz à répartition discrète de vitesses, Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin-New York, 1975 (French). MR 0416379
- Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger, Time dependent problems and difference methods, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1377057
- Michael Herty and Wen-An Yong, Feedback boundary control of linear hyperbolic systems with relaxation, Automatica J. IFAC 69 (2016), 12–17. MR 3500083, DOI 10.1016/j.automatica.2016.02.016
- Robert L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev. 28 (1986), no. 2, 177–217. MR 839822, DOI 10.1137/1028050
- Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, DOI 10.1002/cpa.3160480303
- C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021–1065. MR 1392419, DOI 10.1007/BF02179552
- Hailiang Liu and Wen-An Yong, Time-asymptotic stability of boundary-layers for a hyperbolic relaxation system, Comm. Partial Differential Equations 26 (2001), no. 7-8, 1323–1343. MR 1855280, DOI 10.1081/PDE-100106135
- Andrew Majda and Stanley Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), no. 5, 607–675. MR 410107, DOI 10.1002/cpa.3160280504
- Guy Métivier, Small viscosity and boundary layer methods, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2004. Theory, stability analysis, and applications. MR 2151414, DOI 10.1007/978-0-8176-8214-9
- Luc Mieussens, Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries, J. Comput. Phys. 162 (2000), no. 2, 429–466. MR 1774264, DOI 10.1006/jcph.2000.6548
- Shinya Nishibata, The initial-boundary value problems for hyperbolic conservation laws with relaxation, J. Differential Equations 130 (1996), no. 1, 100–126. MR 1409025, DOI 10.1006/jdeq.1996.0134
- W. Vincenti and C. Jr. Kruger, Introduction to physical gas dynamic, Krieger, Malabar, 1986.
- Wei-Cheng Wang and Zhouping Xin, Asymptotic limit of initial-boundary value problems for conservation laws with relaxational extensions, Comm. Pure Appl. Math. 51 (1998), no. 5, 505–535. MR 1604274, DOI 10.1002/(SICI)1097-0312(199805)51:5$\langle$505::AID-CPA3$\rangle$3.0.CO;2-C
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- Zhouping Xin and Wen-Qing Xu, Stiff well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane, J. Differential Equations 167 (2000), no. 2, 388–437. MR 1793199, DOI 10.1006/jdeq.2000.3806
- Wen-Qing Xu, Boundary conditions and boundary layers for a multi-dimensional relaxation model, J. Differential Equations 197 (2004), no. 1, 85–117. MR 2030150, DOI 10.1016/j.jde.2003.08.007
- Wen-An Yong, Boundary conditions for hyperbolic systems with stiff source terms, Indiana Univ. Math. J. 48 (1999), no. 1, 115–137. MR 1722195, DOI 10.1512/iumj.1999.48.1611
- Wen-An Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations 155 (1999), no. 1, 89–132. MR 1693210, DOI 10.1006/jdeq.1998.3584
- Wen-An Yong, Newtonian limit of Maxwell fluid flows, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 913–922. MR 3269638, DOI 10.1007/s00205-014-0769-2
- Yizhou Zhou and Wen-An Yong, Construction of boundary conditions for hyperbolic relaxation approximations I: The linearized Suliciu model, Math. Models Methods Appl. Sci. 30 (2020), no. 7, 1407–1439. MR 4122254, DOI 10.1142/S0218202520500268
- Yizhou Zhou and Wen-An Yong, Boundary conditions for hyperbolic relaxation systems with characteristic boundaries of type I, J. Differential Equations 281 (2021), 289–332. MR 4213673, DOI 10.1016/j.jde.2021.02.008
- Yizhou Zhou and Wen-An Yong, Boundary conditions for hyperbolic relaxation systems with characteristic boundaries of type II, J. Differential Equations 310 (2022), 198–234. MR 4355918, DOI 10.1016/j.jde.2021.11.020
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
35L50
Retrieve articles in all journals
with MSC (2020):
35L50
Additional Information
Xiaxia Cao
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
ORCID:
0000-0002-7423-4454
Email:
caoxx18@mails.tsinghua.edu.cn
Wen-An Yong
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China —and—Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
Email:
wayong@tsinghua.edu.cn
Received by editor(s):
March 12, 2022
Received by editor(s) in revised form:
April 23, 2022
Published electronically:
May 25, 2022
Additional Notes:
This research was supported by National Natural Science Foundation of China (grant No. 12071246) and by National Key R&D Program of China (grant No. 2021YFA0719200)
Article copyright:
© Copyright 2022
Brown University
