-
Previous Article
Force-based models of pedestrian dynamics
- NHM Home
- This Issue
-
Next Article
On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms
An adaptive finite-volume method for a model of two-phase pedestrian flow
1. | Departamento de Ciencias Matemáticas y Físicas, Universidad Católica de Temuco, Temuco, Chile |
2. | Modeling and Scientific Computing, MATHISCE, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland |
3. | Institut für Mathematik, Fakultät II Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany |
4. | Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB E4L 1G6, Canada |
References:
[1] |
B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math. Models Meth. Appl. Sci., 21 (2011), 307-344.
doi: 10.1142/S0218202511005064. |
[2] |
A. V. Azevedo, D. Marchesin, B. Plohr and K. Zumbrun, Capillary instability in models for three-phase flow, Z. Angew. Math. Phys., 53 (2002), 713-746.
doi: 10.1007/s00033-002-8180-5. |
[3] |
J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flows in porous media, SIAM J. Appl. Math., 46 (1986), 1000-1017.
doi: 10.1137/0146059. |
[4] |
S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. |
[5] |
S. Berres, R. Bürger and K. H. Karlsen, Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions, J. Comput. Appl. Math., 164/165 (2004), 53-80.
doi: 10.1016/S0377-0427(03)00496-5. |
[6] |
S. Berres, R. Bürger and A. Kozakevicius, Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes, Adv. Appl. Math. Mech., 1 (2009), 581-614. |
[7] |
S. Berres and R. Ruiz-Baier, A fully adaptive numerical approximation for a two-dimensional epidemic model with non-linear cross-diffusion, Nonlin. Anal. Real World Appl., 12 (2011), 2888-2903.
doi: 10.1016/j.nonrwa.2011.04.014. |
[8] |
S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, Two-dimensional models of pedestrian flow, in "Series in Contemporary Applied Mathematics" (Proceedings of HYP 2010) (eds. P. G. Ciarlet and Ta-Tsien Li), Higher Education Press, Beijing, World Scientific, Singapore, 2011, to appear. |
[9] |
J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math., 18 (1960), 191-204. |
[10] |
L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445.
doi: 10.1016/j.apm.2010.07.007. |
[11] |
R. Bürger, K. H. Karlsen, E. M. Tory and W. L. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, ZAMM Z. Angew. Math. Mech., 82 (2002), 699-722. |
[12] |
R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media, J. Sci. Comput., 43 (2010), 261-290.
doi: 10.1007/s10915-010-9356-3. |
[13] |
C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Stat. Mech. Appl., 295 (2001), 507-525.
doi: 10.1016/S0378-4371(01)00141-8. |
[14] |
S. Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149. |
[15] |
M. Chen, G. Bärwolff and H. Schwandt, A derived grid-based model for simulation of pedestrian flow, J. Zhejiang Univ.: Science A, 10 (2009), 209-220.
doi: 10.1631/jzus.A0820049. |
[16] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.
doi: 10.1137/100797515. |
[17] |
R. R. Clements and R. L. Hughes, Mathematical modelling of a mediaeval battle: The battle of Agincourt, 1415, Math. Comput. Simul., 64 (2004), 259-269.
doi: 10.1016/j.matcom.2003.09.019. |
[18] |
W. Daamen, P. H. L. Bovy and S. P. Hoogendoorn, Modelling pedestrians in transfer stations, in "Pedestrian and Evacuation Dynamics" (eds. M. Schreckenberg and S. D. Sharma), Springer-Verlag, Berlin, Heidelberg, (2002), 59-73. |
[19] |
J. Esser and M. Schreckenberg, Microscopic simulation of urban traffic based on cellular automata, Int. J. Mod. Phys. C, 8 (1997), 1025-1036.
doi: 10.1142/S0129183197000904. |
[20] |
A. D. Fitt, The numerical and analytical solution of ill-posed systems of conservation laws, Appl. Math. Modelling, 13 (1989), 618-631.
doi: 10.1016/0307-904X(89)90171-6. |
[21] |
H. Frid and I.-S. Liu, Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type, Z. Angew. Math. Phys., 46 (1995), 913-931.
doi: 10.1007/BF00917877. |
[22] |
A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal., 33 (1996), 1205-1256.
doi: 10.1137/0733060. |
[23] |
D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
[24] |
D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24.
doi: 10.1287/trsc.1040.0108. |
[25] |
D. Helbing, I. Farkas and T. Vicsek, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[26] |
H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264.
doi: 10.1002/cpa.3160400206. |
[27] |
S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. |
[28] |
R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.
doi: 10.1016/S0191-2615(01)00015-7. |
[29] |
J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224. |
[30] |
Y. Jiang, P. Zhang, S. C. Wong and R. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635.
doi: 10.1016/j.physa.2010.05.003. |
[31] |
B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83.
doi: 10.1103/PhysRevE.50.54. |
[32] |
B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581. |
[33] |
B. L. Keyfitz, "Mathematical Properties of Nonhyperbolic Models for Incompressible Two-Phase Flow," Proceedings of the International Conference on Multiphase Flow, New Orleans, May 27-June 1, 2001, (CD-ROM). |
[34] |
A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282.
doi: 10.1006/jcph.2000.6459. |
[35] |
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. |
[36] |
A. Majda and R. L. Pego, Stable viscosity matrices for system of conservation laws, J. Diff. Eqns., 56 (1985), 229-262.
doi: 10.1016/0022-0396(85)90107-X. |
[37] |
D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. |
[38] |
S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003. |
[39] |
A. Nakayama, K. Hasebe and Y. Sugiyama, Instability of pedestrian flow in 2D optimal velocity model with attractive interaction, Comput. Phys. Comm., 177 (2007), 162-163.
doi: 10.1016/j.cpc.2007.02.007. |
[40] |
B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
[41] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[42] |
Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. |
[43] |
Th. Slawig, G. Bärwolff and H. Schwandt, "Simulation of Pedestrian Flows for Traffic Control Systems," in Proceedings of the 7th International Conference on Information and Management Sciences (IMS 2008) (Urumtschi, 12. 8. 08 - 19. 8. 08) (ed. E. Y. Li), California Polytechnic State University, Series on Information and Management Sciences, 7, Pomona/California, 2008, 360-374. |
[44] |
E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction," Third edition, Springer-Verlag, Berlin 2009.
doi: 10.1007/b79761. |
[45] |
Y. Xia, S. C. Wong, M. P. Zhang, C.-W. Shu and W. H. K. Lam, An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model, Int. J. Numer. Meth. Engrg., 76 (2008), 337-350.
doi: 10.1002/nme.2329. |
[46] |
P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756.
doi: 10.1016/j.jcp.2005.07.019. |
show all references
References:
[1] |
B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math. Models Meth. Appl. Sci., 21 (2011), 307-344.
doi: 10.1142/S0218202511005064. |
[2] |
A. V. Azevedo, D. Marchesin, B. Plohr and K. Zumbrun, Capillary instability in models for three-phase flow, Z. Angew. Math. Phys., 53 (2002), 713-746.
doi: 10.1007/s00033-002-8180-5. |
[3] |
J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flows in porous media, SIAM J. Appl. Math., 46 (1986), 1000-1017.
doi: 10.1137/0146059. |
[4] |
S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612. |
[5] |
S. Berres, R. Bürger and K. H. Karlsen, Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions, J. Comput. Appl. Math., 164/165 (2004), 53-80.
doi: 10.1016/S0377-0427(03)00496-5. |
[6] |
S. Berres, R. Bürger and A. Kozakevicius, Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes, Adv. Appl. Math. Mech., 1 (2009), 581-614. |
[7] |
S. Berres and R. Ruiz-Baier, A fully adaptive numerical approximation for a two-dimensional epidemic model with non-linear cross-diffusion, Nonlin. Anal. Real World Appl., 12 (2011), 2888-2903.
doi: 10.1016/j.nonrwa.2011.04.014. |
[8] |
S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, Two-dimensional models of pedestrian flow, in "Series in Contemporary Applied Mathematics" (Proceedings of HYP 2010) (eds. P. G. Ciarlet and Ta-Tsien Li), Higher Education Press, Beijing, World Scientific, Singapore, 2011, to appear. |
[9] |
J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math., 18 (1960), 191-204. |
[10] |
L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445.
doi: 10.1016/j.apm.2010.07.007. |
[11] |
R. Bürger, K. H. Karlsen, E. M. Tory and W. L. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, ZAMM Z. Angew. Math. Mech., 82 (2002), 699-722. |
[12] |
R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media, J. Sci. Comput., 43 (2010), 261-290.
doi: 10.1007/s10915-010-9356-3. |
[13] |
C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Stat. Mech. Appl., 295 (2001), 507-525.
doi: 10.1016/S0378-4371(01)00141-8. |
[14] |
S. Čanić, On the influence of viscosity on Riemann solutions, J. Dyn. Diff. Eqns., 10 (1998), 109-149. |
[15] |
M. Chen, G. Bärwolff and H. Schwandt, A derived grid-based model for simulation of pedestrian flow, J. Zhejiang Univ.: Science A, 10 (2009), 209-220.
doi: 10.1631/jzus.A0820049. |
[16] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.
doi: 10.1137/100797515. |
[17] |
R. R. Clements and R. L. Hughes, Mathematical modelling of a mediaeval battle: The battle of Agincourt, 1415, Math. Comput. Simul., 64 (2004), 259-269.
doi: 10.1016/j.matcom.2003.09.019. |
[18] |
W. Daamen, P. H. L. Bovy and S. P. Hoogendoorn, Modelling pedestrians in transfer stations, in "Pedestrian and Evacuation Dynamics" (eds. M. Schreckenberg and S. D. Sharma), Springer-Verlag, Berlin, Heidelberg, (2002), 59-73. |
[19] |
J. Esser and M. Schreckenberg, Microscopic simulation of urban traffic based on cellular automata, Int. J. Mod. Phys. C, 8 (1997), 1025-1036.
doi: 10.1142/S0129183197000904. |
[20] |
A. D. Fitt, The numerical and analytical solution of ill-posed systems of conservation laws, Appl. Math. Modelling, 13 (1989), 618-631.
doi: 10.1016/0307-904X(89)90171-6. |
[21] |
H. Frid and I.-S. Liu, Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type, Z. Angew. Math. Phys., 46 (1995), 913-931.
doi: 10.1007/BF00917877. |
[22] |
A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal., 33 (1996), 1205-1256.
doi: 10.1137/0733060. |
[23] |
D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
[24] |
D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24.
doi: 10.1287/trsc.1040.0108. |
[25] |
D. Helbing, I. Farkas and T. Vicsek, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[26] |
H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264.
doi: 10.1002/cpa.3160400206. |
[27] |
S. P. Hoogendoorn and W. Daamen, Self-organization in pedestrian flow, Traff. Granul. Flow, 3 (2005), 373-382. |
[28] |
R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.
doi: 10.1016/S0191-2615(01)00015-7. |
[29] |
J. Hurley and B. J. Plohr, Some effects of viscous terms on Riemann Problem solutions, Math. Contemp., 8 (1995), 203-224. |
[30] |
Y. Jiang, P. Zhang, S. C. Wong and R. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635.
doi: 10.1016/j.physa.2010.05.003. |
[31] |
B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83.
doi: 10.1103/PhysRevE.50.54. |
[32] |
B. L. Keyfitz, A geometric theory of conservation laws which change type, ZAMM Z. Angew. Math. Mech., 75 (1995), 571-581. |
[33] |
B. L. Keyfitz, "Mathematical Properties of Nonhyperbolic Models for Incompressible Two-Phase Flow," Proceedings of the International Conference on Multiphase Flow, New Orleans, May 27-June 1, 2001, (CD-ROM). |
[34] |
A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282.
doi: 10.1006/jcph.2000.6459. |
[35] |
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. |
[36] |
A. Majda and R. L. Pego, Stable viscosity matrices for system of conservation laws, J. Diff. Eqns., 56 (1985), 229-262.
doi: 10.1016/0022-0396(85)90107-X. |
[37] |
D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE, paper 56480. |
[38] |
S. Müller, "Adaptive Multiscale Schemes for Conservation Laws," Lecture Notes in Computational Science and Engineering, 27, Springer-Verlag, Berlin, 2003. |
[39] |
A. Nakayama, K. Hasebe and Y. Sugiyama, Instability of pedestrian flow in 2D optimal velocity model with attractive interaction, Comput. Phys. Comm., 177 (2007), 162-163.
doi: 10.1016/j.cpc.2007.02.007. |
[40] |
B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
[41] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[42] |
Th. Slawig, G. Bärwolff and H. Schwandt, "Hybrid macro-Microscopic Simulation of Pedestrian Flow," Technical report, Institut für Mathematik, Technische Universität Berlin, 2007. |
[43] |
Th. Slawig, G. Bärwolff and H. Schwandt, "Simulation of Pedestrian Flows for Traffic Control Systems," in Proceedings of the 7th International Conference on Information and Management Sciences (IMS 2008) (Urumtschi, 12. 8. 08 - 19. 8. 08) (ed. E. Y. Li), California Polytechnic State University, Series on Information and Management Sciences, 7, Pomona/California, 2008, 360-374. |
[44] |
E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction," Third edition, Springer-Verlag, Berlin 2009.
doi: 10.1007/b79761. |
[45] |
Y. Xia, S. C. Wong, M. P. Zhang, C.-W. Shu and W. H. K. Lam, An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model, Int. J. Numer. Meth. Engrg., 76 (2008), 337-350.
doi: 10.1002/nme.2329. |
[46] |
P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756.
doi: 10.1016/j.jcp.2005.07.019. |
[1] |
Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329 |
[2] |
Steinar Evje, Kenneth H. Karlsen. Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1 (4) : 639-673. doi: 10.3934/nhm.2006.1.639 |
[3] |
N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079 |
[4] |
Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks and Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749 |
[5] |
Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045 |
[6] |
Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 |
[7] |
Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143 |
[8] |
Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253 |
[9] |
Lucas C. F. Ferreira, Everaldo Medeiros, Marcelo Montenegro. An elliptic system and the critical hyperbola. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1169-1182. doi: 10.3934/cpaa.2015.14.1169 |
[10] |
K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091 |
[11] |
Hongwei Lou, Xueyuan Yin. Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors. Mathematical Control and Related Fields, 2018, 8 (3&4) : 855-877. doi: 10.3934/mcrf.2018038 |
[12] |
Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 |
[13] |
Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111 |
[14] |
Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673 |
[15] |
Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011 |
[16] |
Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51 |
[17] |
Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717 |
[18] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
[19] |
Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15 |
[20] |
Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]