June  2018, 11(3): 491-520. doi: 10.3934/krm.2018022

Kinetic description of collision avoidance in pedestrian crowds by sidestepping

1. 

RICAM, Austrian Academy of Sciences (ÖAW), Altenbergerstr. 69, 4040 Linz, Austria

2. 

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy

3. 

Mathematical Institute, University of Warwick, CV4 7AL Coventry, UK

4. 

RICAM, Austrian Academy of Sciences (ÖAW), Altenbergerstr. 69, 4040 Linz, Austria

* Corresponding author

Received  October 2016 Revised  July 2017 Published  March 2018

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

Citation: Adriano Festa, Andrea Tosin, Marie-Therese Wolfram. Kinetic description of collision avoidance in pedestrian crowds by sidestepping. Kinetic & Related Models, 2018, 11 (3) : 491-520. doi: 10.3934/krm.2018022
References:
[1]

J. P. AgnelliF. Colasuonno and D. Knopoff, A kinetic theory approach to the dynamics of crowd evacuation from bounded domains, Math. Models Methods Appl. Sci., 25 (2015), 109-129. doi: 10.1142/S0218202515500049. Google Scholar

[2]

G. AlbiM. BonginiE. Cristiani and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710. doi: 10.1137/15M1017016. Google Scholar

[3]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar

[4]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Phil. Trans. R. Soc. A, 372 (2014), 20140138, 18 pp. Google Scholar

[5]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14 pp. Google Scholar

[6]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar

[7]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-4586. doi: 10.1103/PhysRevE.61.4576. Google Scholar

[8]

R. BorscheR.M. ColomboM. Garavello and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015), 827-859. doi: 10.1007/s00332-015-9242-0. Google Scholar

[9]

M. BurgerS. HittmeirH. Ranetbauer and M.-T. Wolfram, Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005. doi: 10.1137/15M1033174. Google Scholar

[10]

C. BursteddeK. KlauckA. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Phys. A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[11]

C. CanutoF. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243-265. doi: 10.1137/100793177. Google Scholar

[12]

E. CarliniA. FestaF. J. Silva and M.-T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes' model for pedestrian flow, Dyn. Games Appl., 7 (2017), 683-705. doi: 10.1007/s13235-016-0202-6. Google Scholar

[13]

E. Carlini and F. J. Silva, A Semi-Lagrangian scheme for the Fokker-Planck equation, IFAC-PapersOnLine, 49 (2016), 272-277. doi: 10.1016/j.ifacol.2016.07.453. Google Scholar

[14]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[15]

A. ColombiM. Scianna and A. Tosin, Moving in a crowd: Human perception as a multiscale process, J. Coupled Syst. Multiscale Dyn., 4 (2016), 25-29. doi: 10.1166/jcsmd.2016.1093. Google Scholar

[16]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023 (34 pages). Google Scholar

[17]

E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 2010,337-364. Google Scholar

[18]

E. CristianiB. 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. Google Scholar

[19]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. Google Scholar

[20]

N. CrouseillesM. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations, J. Comput. Phys., 229 (2010), 1927-1953. doi: 10.1016/j.jcp.2009.11.007. Google Scholar

[21]

P. DegondC. Appert-RollandJ. Pettré and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809. Google Scholar

[22]

A. Festa and M. -T. Wolfram, Collision avoidance in pedestrian dynamics, in Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015,3187-3192. doi: 10.1109/CDC.2015.7402697. Google Scholar

[23]

D. HelbingL. BuznaA. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transport. Sci., 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108. Google Scholar

[24]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[25]

S. Hoogendoorn and P. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transport. Res. B-Meth., 38 (2004), 169-190. doi: 10.1016/S0191-2615(03)00007-9. Google Scholar

[26]

C.-S. HuangT. Arbogast and C.-H. Hung, A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws, J. Comput. Phys., 322 (2016), 559-585. doi: 10.1016/j.jcp.2016.06.027. Google Scholar

[27]

I. Karamouzas, B. Skinner and S. J. Guy, Universal power law governing pedestrian interactions, Phys. Rew. Lett., 113 (2014), 238701. doi: 10.1103/PhysRevLett.113.238701. Google Scholar

[28]

A. Klar and R. Wegener, Kinetic traffic flow models, in Modeling in Applied Sciences: A Kinetic Theory Approach (eds. N. Bellomo and M. Pulvirenti), Birkhäuser Boston, 2000, chapter 8,263-316. Google Scholar

[29]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[30]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS One, 5 (2010), e10047.Google Scholar

[31]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2013.Google Scholar

[32]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: Convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105. doi: 10.1007/s10440-012-9771-6. Google Scholar

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science (ed. R. A. Meyers), Springer, New York, 1/2 (2012), 1748-1770. Google Scholar

[34]

A. PolusJ. Schofer and A. Ushpiz, Pedestrian flow and level of service, J. Transp. Eng., 109 (1983), 46-56. Google Scholar

[35]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Netw. Heterog. Media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545. Google Scholar

[36]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[37]

A. Tosin and P. Frasca, Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heterog. Media, 6 (2011), 561-596. doi: 10.3934/nhm.2011.6.561. Google Scholar

[38]

M. TwarogowskaP. Goatin and R. Duvigneau, Macroscopic modelling and simulations of room evacuation, Appl. Math. Model., 38 (2014), 5781-5795. doi: 10.1016/j.apm.2014.03.027. Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics (eds. S. Friedlander and D. Serre), vol. Ⅰ, Elsevier, 2002, 71-305. Google Scholar

[40]

U. Weidmann, Transporttechnik der Fussgänger, Technical report, ETH, Zürich, 1992.Google Scholar

[41]

J. Zhang and A. Seyfried, Comparison of intersecting pedestrian flows based on experiments, Phys. A, 405 (2014), 316-325. doi: 10.1016/j.physa.2014.03.004. Google Scholar

show all references

References:
[1]

J. P. AgnelliF. Colasuonno and D. Knopoff, A kinetic theory approach to the dynamics of crowd evacuation from bounded domains, Math. Models Methods Appl. Sci., 25 (2015), 109-129. doi: 10.1142/S0218202515500049. Google Scholar

[2]

G. AlbiM. BonginiE. Cristiani and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710. doi: 10.1137/15M1017016. Google Scholar

[3]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar

[4]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Phil. Trans. R. Soc. A, 372 (2014), 20140138, 18 pp. Google Scholar

[5]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14 pp. Google Scholar

[6]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar

[7]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-4586. doi: 10.1103/PhysRevE.61.4576. Google Scholar

[8]

R. BorscheR.M. ColomboM. Garavello and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015), 827-859. doi: 10.1007/s00332-015-9242-0. Google Scholar

[9]

M. BurgerS. HittmeirH. Ranetbauer and M.-T. Wolfram, Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005. doi: 10.1137/15M1033174. Google Scholar

[10]

C. BursteddeK. KlauckA. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Phys. A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[11]

C. CanutoF. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243-265. doi: 10.1137/100793177. Google Scholar

[12]

E. CarliniA. FestaF. J. Silva and M.-T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes' model for pedestrian flow, Dyn. Games Appl., 7 (2017), 683-705. doi: 10.1007/s13235-016-0202-6. Google Scholar

[13]

E. Carlini and F. J. Silva, A Semi-Lagrangian scheme for the Fokker-Planck equation, IFAC-PapersOnLine, 49 (2016), 272-277. doi: 10.1016/j.ifacol.2016.07.453. Google Scholar

[14]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[15]

A. ColombiM. Scianna and A. Tosin, Moving in a crowd: Human perception as a multiscale process, J. Coupled Syst. Multiscale Dyn., 4 (2016), 25-29. doi: 10.1166/jcsmd.2016.1093. Google Scholar

[16]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023 (34 pages). Google Scholar

[17]

E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 2010,337-364. Google Scholar

[18]

E. CristianiB. 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. Google Scholar

[19]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. Google Scholar

[20]

N. CrouseillesM. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations, J. Comput. Phys., 229 (2010), 1927-1953. doi: 10.1016/j.jcp.2009.11.007. Google Scholar

[21]

P. DegondC. Appert-RollandJ. Pettré and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809. Google Scholar

[22]

A. Festa and M. -T. Wolfram, Collision avoidance in pedestrian dynamics, in Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015,3187-3192. doi: 10.1109/CDC.2015.7402697. Google Scholar

[23]

D. HelbingL. BuznaA. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transport. Sci., 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108. Google Scholar

[24]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[25]

S. Hoogendoorn and P. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transport. Res. B-Meth., 38 (2004), 169-190. doi: 10.1016/S0191-2615(03)00007-9. Google Scholar

[26]

C.-S. HuangT. Arbogast and C.-H. Hung, A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws, J. Comput. Phys., 322 (2016), 559-585. doi: 10.1016/j.jcp.2016.06.027. Google Scholar

[27]

I. Karamouzas, B. Skinner and S. J. Guy, Universal power law governing pedestrian interactions, Phys. Rew. Lett., 113 (2014), 238701. doi: 10.1103/PhysRevLett.113.238701. Google Scholar

[28]

A. Klar and R. Wegener, Kinetic traffic flow models, in Modeling in Applied Sciences: A Kinetic Theory Approach (eds. N. Bellomo and M. Pulvirenti), Birkhäuser Boston, 2000, chapter 8,263-316. Google Scholar

[29]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[30]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS One, 5 (2010), e10047.Google Scholar

[31]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2013.Google Scholar

[32]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: Convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105. doi: 10.1007/s10440-012-9771-6. Google Scholar

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science (ed. R. A. Meyers), Springer, New York, 1/2 (2012), 1748-1770. Google Scholar

[34]

A. PolusJ. Schofer and A. Ushpiz, Pedestrian flow and level of service, J. Transp. Eng., 109 (1983), 46-56. Google Scholar

[35]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Netw. Heterog. Media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545. Google Scholar

[36]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[37]

A. Tosin and P. Frasca, Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heterog. Media, 6 (2011), 561-596. doi: 10.3934/nhm.2011.6.561. Google Scholar

[38]

M. TwarogowskaP. Goatin and R. Duvigneau, Macroscopic modelling and simulations of room evacuation, Appl. Math. Model., 38 (2014), 5781-5795. doi: 10.1016/j.apm.2014.03.027. Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics (eds. S. Friedlander and D. Serre), vol. Ⅰ, Elsevier, 2002, 71-305. Google Scholar

[40]

U. Weidmann, Transporttechnik der Fussgänger, Technical report, ETH, Zürich, 1992.Google Scholar

[41]

J. Zhang and A. Seyfried, Comparison of intersecting pedestrian flows based on experiments, Phys. A, 405 (2014), 316-325. doi: 10.1016/j.physa.2014.03.004. Google Scholar

Figure 1.  Calculation of the time to collision.
Figure 2.  The state variables describing pedestrians moving with constant speed with $\alpha_d = 0$.
Figure 3.  Comparison between (5) (blue curves computed with $\gamma = 0.8$, $\tau = 1$) and (6) (red curves). Left: $\theta_j = \pi$; right: $\theta_j = -\frac{\pi}{4}$ (see the text for details).
Figure 4.  The set of pairs $(\rho,\,|\alpha_c|)\in [0,\,1]\times [0,\,\pi]$ (shaded area) for which Theorem 4.1 guarantees alignment in the case study of Section 4.2.
Figure 5.  Alignment to the desired direction $\alpha_d = 0$ for the case study of Section 4.2 with $\rho = \frac{1}{2}$, $\alpha_c = \frac{\pi}{5}$. Left: $t = 10^{-2}$, right: $t = 10$.
Figure 6.  The mapping $t\mapsto\bar{\theta}(t)$ computed numerically (blue solid curve) and estimated theoretically by (19) (red dashed curve). Left: $(\rho,\,\alpha_c) = \left(\frac{1}{2},\,\frac{\pi}{5}\right)$; right: $(\rho,\,\alpha_c) = \left(\frac{1}{3},\,\frac{3}{5}\pi\right)$.
Figure 7.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{MC}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (centre) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$; (right) $\rho = 1$, $\alpha_c = \frac{2}{5}\pi$ and different values of $\bar{\theta}_0$ obtained by varying the parameter $\hat{\theta}$ in (22). Dashed curves indicate the cases in which alignment fails.
Figure 8.  Conditions (14) (blue) with $\bar{\theta}_0 = \frac{\pi}{2}$ and (31) (red) for two different choices of $a(\rho)$. Purple areas correspond to the pairs $(\rho,\,|\alpha_c|)$ satisfying both conditions. The values $\rho_1$, $\rho_2$ in the right graph are $\rho_{1,2} = \frac{21\mp\sqrt{105}}{42}$, i.e. $\rho_1\approx 0.256$, $\rho_2\approx 0.744$.
Figure 9.  Top left: snapshots in time of the solution to (27) with initial condition (20). Top right: the same solution plotted in the $\theta t$-plane. Bottom: snapshots in time of the solution to (27) with initial condition (22) for two different choices of $\hat{\theta}$, $\sigma^2$.
Figure 10.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{FEM}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (right) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$. Alignment is always observed.
Figure 11.  Test 1 -Alignment in the spatially inhomogeneous case and time trend of ${{\mathit{\bar{\theta }}}_{\rm{MC}}}$ over the whole space domain $\Omega$.
Figure 12.  Test 2 -Lane formation in pedestrian counterflow. For pictorial purposes, in order to represent the two crowds in the same picture, the blue density is plotted on a negative scale. In white areas where the velocity is non-zero the two densities take nearly the same values.
Figure 13.  Test 3 -Crossing flows and stripe formation.
Table 1.  Parameters used in the numerical simulations of Section 5.
$L$ $\alpha_d$ $\alpha_d^1$ $\alpha_d^2$ $\alpha_c$ $\tau$ $\gamma$ $\eta$
Test 1 10 $\pi$ - - $\frac{\pi}{4}$ 1 $\frac{1}{2}$ -
Test 2 10 - 0 $-\pi$ $\frac{\pi}{4}$ 1 $\frac{1}{2}$ $\frac{1}{5}$
Test 3 10 - 0 $\frac{\pi}{2}$ $\frac{\pi}{4}$ 10 $\frac{2}{5}$ $\frac{1}{5}$
$L$ $\alpha_d$ $\alpha_d^1$ $\alpha_d^2$ $\alpha_c$ $\tau$ $\gamma$ $\eta$
Test 1 10 $\pi$ - - $\frac{\pi}{4}$ 1 $\frac{1}{2}$ -
Test 2 10 - 0 $-\pi$ $\frac{\pi}{4}$ 1 $\frac{1}{2}$ $\frac{1}{5}$
Test 3 10 - 0 $\frac{\pi}{2}$ $\frac{\pi}{4}$ 10 $\frac{2}{5}$ $\frac{1}{5}$
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