December  2017, 12(4): 591-617. doi: 10.3934/nhm.2017024

Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

1. 

Université Côte d'Azur, Inria, CNRS, LJAD, Parc Valrose, 06108 Nice, France

2. 

Institut de Mécanique des Fluides de Toulouse, CNRS UMR 5502, France

3. 

Université d'Orléans, MAPMO, UMR CNRS 7349, France

Received  December 2016 Revised  August 2017 Published  October 2017

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

Citation: Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024
References:
[1]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278, doi: 10.1137/S0036139900380955.  Google Scholar

[2]

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.  Google Scholar

[3]

N. BellomoA. BellouquidJ. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Disc. Cont. Dyn. Syst.-B, 19 (2014), 1869-1888.  doi: 10.3934/dcdsb.2014.19.1869.  Google Scholar

[4]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677.  Google Scholar

[5]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford Mathematical Monographs, Oxford University Press, 2007.  Google Scholar

[6]

F. Berthelin and D. Broizat, A model for the evolution of traffic jams in multilane, Kinetic and Related Models, 5 (2012), 697-728.  doi: 10.3934/krm.2012.5.697.  Google Scholar

[7]

F. BerthelinT. Goudon and S. Minjeaud, Multifluid flows: A kinetic approach, J. Sci. Comput., 66 (2016), 792-824.  doi: 10.1007/s10915-015-0044-1.  Google Scholar

[8]

F. BerthelinP. DegondM. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[9]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in math., Birkhäuser, 2004. doi: 10.1007/b93802.  Google Scholar

[10]

F. BouchutS. Jin and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), 135-158 (electronic).  doi: 10.1137/S0036142901398040.  Google Scholar

[11]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.  doi: 10.1137/S0036142997317353.  Google Scholar

[12]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[13]

C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221.  doi: 10.4171/IFB/186.  Google Scholar

[14]

P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110.  doi: 10.1137/0903007.  Google Scholar

[15]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[16]

P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293.  doi: 10.3934/krm.2008.1.279.  Google Scholar

[17]

P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31.  doi: 10.4208/cicp.210709.210610a.  Google Scholar

[18]

P. Degond and J. Hua, Self-organized hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319.  doi: 10.1016/j.jcp.2012.11.033.  Google Scholar

[19]

P. DegondJ. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088.  doi: 10.1016/j.jcp.2011.07.010.  Google Scholar

[20]

D. C. GazisR. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Research, 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[21]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Applied Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar

[22]

E. Grenier, Existence globale pour le systéme des gaz sans pression, Comptes Rendus Acad. Sci., 321 (1995), 171-174.   Google Scholar

[23]

J. Jung, Schémas numériques adaptés aux accélérateurs multicoeurs pour les écoulements bifluides, PhD thesis, Univ. Strasbourg, 2014. Google Scholar

[24]

A.-Y. Le Roux, Stability for some equations of gas dynamics, Math. Comput., 37 (1981), 307-320.  doi: 10.1090/S0025-5718-1981-0628697-8.  Google Scholar

[25]

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.  Google Scholar

[26]

P. -L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 373-410, doi: 10.1016/S0294-1449(99)80018-3.  Google Scholar

[27]

T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 57 (1977), 135-148.  doi: 10.1007/BF01625772.  Google Scholar

[28]

B. Maury and A. Preux, Pressureless Euler equations with maximal density constraint: A time-splitting scheme, Technical report, Université Paris-Sud, 2017, 333-355, Available on https://hal.archives-ouvertes.fr/hal-01224008. doi: 10.1515/9783110430417-014.  Google Scholar

[29]

P. Nelson and A. Sopasakis, The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations, Transportation Research Part B: Methodological, 32 (1998), 589-604.  doi: 10.1016/S0191-2615(98)00020-4.  Google Scholar

[30]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research, 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[31]

H. J. Payne, Freflo: A Macroscopic Simulation Model of Freeway Traffic, Transportation Research Record. Google Scholar

[32]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing, 1971. Google Scholar

[33]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Communications in Mathematical Sciences, 14 (2016), 643-669.  doi: 10.4310/CMS.2016.v14.n3.a3.  Google Scholar

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[36]

R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport Theory and Stat. Phys., 25 (1996), 785-798.  doi: 10.1080/00411459608203547.  Google Scholar

[37]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

show all references

References:
[1]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278, doi: 10.1137/S0036139900380955.  Google Scholar

[2]

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.  Google Scholar

[3]

N. BellomoA. BellouquidJ. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Disc. Cont. Dyn. Syst.-B, 19 (2014), 1869-1888.  doi: 10.3934/dcdsb.2014.19.1869.  Google Scholar

[4]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677.  Google Scholar

[5]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford Mathematical Monographs, Oxford University Press, 2007.  Google Scholar

[6]

F. Berthelin and D. Broizat, A model for the evolution of traffic jams in multilane, Kinetic and Related Models, 5 (2012), 697-728.  doi: 10.3934/krm.2012.5.697.  Google Scholar

[7]

F. BerthelinT. Goudon and S. Minjeaud, Multifluid flows: A kinetic approach, J. Sci. Comput., 66 (2016), 792-824.  doi: 10.1007/s10915-015-0044-1.  Google Scholar

[8]

F. BerthelinP. DegondM. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.  Google Scholar

[9]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in math., Birkhäuser, 2004. doi: 10.1007/b93802.  Google Scholar

[10]

F. BouchutS. Jin and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), 135-158 (electronic).  doi: 10.1137/S0036142901398040.  Google Scholar

[11]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.  doi: 10.1137/S0036142997317353.  Google Scholar

[12]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[13]

C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221.  doi: 10.4171/IFB/186.  Google Scholar

[14]

P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110.  doi: 10.1137/0903007.  Google Scholar

[15]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[16]

P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293.  doi: 10.3934/krm.2008.1.279.  Google Scholar

[17]

P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31.  doi: 10.4208/cicp.210709.210610a.  Google Scholar

[18]

P. Degond and J. Hua, Self-organized hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319.  doi: 10.1016/j.jcp.2012.11.033.  Google Scholar

[19]

P. DegondJ. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088.  doi: 10.1016/j.jcp.2011.07.010.  Google Scholar

[20]

D. C. GazisR. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Research, 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[21]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Applied Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar

[22]

E. Grenier, Existence globale pour le systéme des gaz sans pression, Comptes Rendus Acad. Sci., 321 (1995), 171-174.   Google Scholar

[23]

J. Jung, Schémas numériques adaptés aux accélérateurs multicoeurs pour les écoulements bifluides, PhD thesis, Univ. Strasbourg, 2014. Google Scholar

[24]

A.-Y. Le Roux, Stability for some equations of gas dynamics, Math. Comput., 37 (1981), 307-320.  doi: 10.1090/S0025-5718-1981-0628697-8.  Google Scholar

[25]

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.  Google Scholar

[26]

P. -L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 373-410, doi: 10.1016/S0294-1449(99)80018-3.  Google Scholar

[27]

T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 57 (1977), 135-148.  doi: 10.1007/BF01625772.  Google Scholar

[28]

B. Maury and A. Preux, Pressureless Euler equations with maximal density constraint: A time-splitting scheme, Technical report, Université Paris-Sud, 2017, 333-355, Available on https://hal.archives-ouvertes.fr/hal-01224008. doi: 10.1515/9783110430417-014.  Google Scholar

[29]

P. Nelson and A. Sopasakis, The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations, Transportation Research Part B: Methodological, 32 (1998), 589-604.  doi: 10.1016/S0191-2615(98)00020-4.  Google Scholar

[30]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research, 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[31]

H. J. Payne, Freflo: A Macroscopic Simulation Model of Freeway Traffic, Transportation Research Record. Google Scholar

[32]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing, 1971. Google Scholar

[33]

G. PuppoM. SempliceA. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Communications in Mathematical Sciences, 14 (2016), 643-669.  doi: 10.4310/CMS.2016.v14.n3.a3.  Google Scholar

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[36]

R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport Theory and Stat. Phys., 25 (1996), 785-798.  doi: 10.1080/00411459608203547.  Google Scholar

[37]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

Figure 1.  Numerical results in the case of transport (15) - (16). Density (left) and velocity (right) with the Glimm scheme (top) and the explicit-implicit scheme (bottom). The results are given for the three different pressures under consideration: pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Figure 2.  Numerical results in the case of decongestion (17) - (18). Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.2$. Top: pressure (VO1) with Glimm scheme (black). On the top, the simulations are performed implicit-explicit scheme (blue) and pressure (VO2) with implicit-explicit scheme (green). Bottom: pressure (VO3) with Glimm scheme (black) and with the explicit-implicit scheme (blue). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Figure 3.  Numerical results in the case of decongestion (17) - (18). Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: $\gamma=4$ (black), $\gamma=20$ (blue) and $\gamma=100$ (green). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Figure 4.  Numerical results in the case of decongestion (17) - (18). Pressure (VO2) for different values of $\varepsilon $ and $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: (VO2) for $\gamma=2, \varepsilon =10^{-3}$ (black), $\gamma=2, \varepsilon =10^{-5}$ (blue), $\gamma=3, \varepsilon =10^{-5}$ (green) and $\gamma=3, \varepsilon =10^{-7}$ (orange). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Figure 5.  Numerical results in the case of congestion -Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.01$. Glimm scheme (top) and implicit-explicit scheme (bottom), with pressure (VO1) (black), (VO2) (blue) and (VO3) (green). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Figure 6.  Numerical results in the case of congestion -Comparison of the two schemes -Different parameters. For (VO1) and (VO2), we use $\gamma=2$ and $\varepsilon =10^{-5}$; for (VO3), we take $\gamma=50$. Pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Figure 7.  Numerical results in the case of congestion -Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.01$. Pressure (VO3) for $\gamma=20$ (black), $\gamma=50$ (blue) and $\gamma=100$ (green). The simulations are performed with the Glimm scheme (top) and the mplicit-explicit scheme (bottom). The exact solution is plotted in red
Figure 8.  Numerical results in the case of congestion -Pressure (VO2) for different values of $\gamma$ and $\varepsilon $. Density (left) and velocity (right) at final time $T=0.01$. We compare the pressure (VO2) for $\gamma=2, \varepsilon =10^{-5}$ (black), $\gamma=3, \varepsilon =10^{-5}$ (blue) and $\gamma=3, \varepsilon =10^{-7}$ (green). The simulations are performed with the Glimm scheme (top) and the implicit-explicit scheme (bottom). The exact solution is plotted in red
Figure 9.  Numerical results in the case of a shock between two blocks -Pressure (VO1) for $\gamma=2$ and different values of $\varepsilon $: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). Figure 9a (on the left) represents the densities whereas figure 9b (on the right) represents the velocities. The exact solution is plotted in red
Figure 10.  Numerical results in the case of a shock between two blocks -Pressure VO2 for $\gamma=2$ and different values of $\varepsilon $: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Figure 11.  Numerical results in the case of a shock between two blocks -Pressure VO3 for different values of $\gamma$: $\gamma=16$ (black), $\gamma=32$ (blue), $\gamma=64$ (green) and $\gamma=128$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Figure 12.  Comparaison of the numerical scheme and velocity offset for the initial data (AI). We display the densities on the left and the velocities on the right, at $t=0.2$ (top), $t=0.4$ (middle) and $t=0.6$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Figure 13.  Comparaison of the numerical scheme and velocity offset for the initial data (AⅢ). We display the densities on the left and the velocities on the right, at $t=0.27$ (top), $t=0.53$ (middle) and $t=0.8$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Table 1.  Time steps -comparison between Glimm scheme and the explicit-implicit scheme for the congestion case. Pressure (VO3) for different values of γ and pressure (VO2) for γ = 2 and different values of ". The time step is the smallest time step used during the simulation and the factor is the ratio of the time step for the explicit-implicit scheme over the time step for the Glimm scheme
Pressure
& param
Time step
Glimm scheme
Time step
explicit-implicit scheme
Factor
Pressure (VO2), ε = 10−4t = 2·10−6t = 2·10−61
Pressure (VO2), ε = 10−5t = 7·10−7t = 10−61.39
Pressure (VO2), ε = 10−6t = 2·10−7t = 7:7·10−73.22
Pressure (VO2), ε = 10−7t = 7:5·10−8t = 6:2·10−78.18
Pressure (VO3), γ = 50t = 9·10−6t = 10−51.12
Pressure (VO3), γ = 100t = 4:8·10−6t = 6:4·10−61.36
Pressure (VO3), γ = 200t = 2:4·10−6t = 5:6·10−62.33
Pressure (VO3), γ = 500t = 9:5·10−7t = 2:7·10−527.95
Pressure
& param
Time step
Glimm scheme
Time step
explicit-implicit scheme
Factor
Pressure (VO2), ε = 10−4t = 2·10−6t = 2·10−61
Pressure (VO2), ε = 10−5t = 7·10−7t = 10−61.39
Pressure (VO2), ε = 10−6t = 2·10−7t = 7:7·10−73.22
Pressure (VO2), ε = 10−7t = 7:5·10−8t = 6:2·10−78.18
Pressure (VO3), γ = 50t = 9·10−6t = 10−51.12
Pressure (VO3), γ = 100t = 4:8·10−6t = 6:4·10−61.36
Pressure (VO3), γ = 200t = 2:4·10−6t = 5:6·10−62.33
Pressure (VO3), γ = 500t = 9:5·10−7t = 2:7·10−527.95
[1]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146

[2]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195

[3]

Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic & Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026

[4]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[5]

Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286

[6]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093

[7]

Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495

[8]

Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153

[9]

Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34

[10]

Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015

[11]

Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431

[12]

Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks & Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1

[13]

Paolo Aluffi. Segre classes of monomial schemes. Electronic Research Announcements, 2013, 20: 55-70. doi: 10.3934/era.2013.20.55

[14]

Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure & Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761

[15]

Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229

[16]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229

[17]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[18]

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

[19]

Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415

[20]

Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (28)
  • HTML views (149)
  • Cited by (0)

[Back to Top]