Efficient numerical methods for multiscale crowd dynamics with emotional contagion

In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [A. Bertozzi, J. Rosado, M. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys. 158 (2014) 647–664]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations — using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the efficiency of these new methods.

[1]  Guillaume Bal,et al.  Mathematical Modelling and Numerical Analysis Coupling of Transport and Diffusion Models in Linear Transport Theory , 2022 .

[2]  Stanley Osher,et al.  Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation , 2005 .

[3]  Tibor Bosse,et al.  Multi-Agent Model For Mutual Absorption Of Emotions , 2009, ECMS.

[4]  Massimo Fornasier,et al.  Particle, kinetic, and hydrodynamic models of swarming , 2010 .

[5]  Tong Zhang,et al.  The Riemann Problem for the Transportation Equations in Gas Dynamics , 1999 .

[6]  Jiequan Li,et al.  Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics , 2001 .

[7]  Tong Zhang,et al.  Generalized Rankine-Hugoniot Relations of Delta-shocks in Solutions of Transportation Equations , 1998 .

[8]  Martin Burger,et al.  Partial differential equation models in the socio-economic sciences , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  Pierre Degond,et al.  A hybrid kinetic/fluid model for solving the gas dynamics Boltzmann-BGK equation , 2004 .

[10]  R. LeVeque Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[11]  F. Bouchut ON ZERO PRESSURE GAS DYNAMICS , 1996 .

[12]  Hailiang Liu,et al.  Formation of δ-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids , 2003, SIAM J. Math. Anal..

[13]  Francis Filbet,et al.  A Hierarchy of Hybrid Numerical Methods for Multiscale Kinetic Equations , 2014, SIAM J. Sci. Comput..

[14]  Patrick Le Tallec,et al.  Coupling Boltzmann and Navier-Stokes Equations by Half Fluxes , 1997 .

[15]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[16]  B. Engquist,et al.  Multi-phase computations in geometrical optics , 1996 .

[17]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[18]  Nicola Bellomo,et al.  From the Microscale to Collective Crowd Dynamics , 2013, Multiscale Model. Simul..

[19]  Axel Klar,et al.  Transition from Kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms , 2000 .

[20]  Eitan Tadmor,et al.  Critical thresholds in 1D Euler equations with nonlocal forces , 2014, 1411.1791.

[21]  R. Hughes The flow of human crowds , 2003 .

[22]  Milind Tambe,et al.  Empirical Evaluation of Computational Emotional Contagion Models , 2011, IVA.

[23]  Patrick Le Tallec,et al.  Coupling Boltzmann and Euler equations without overlapping , 1992 .

[24]  R. LeVeque Numerical methods for conservation laws , 1990 .

[25]  Li Wang,et al.  A domain decomposition method for semilinear hyperbolic systems with two-scale relaxations , 2012, Math. Comput..

[26]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[27]  Giacomo Dimarco,et al.  Hybrid Multiscale Methods II. Kinetic Equations , 2008, Multiscale Model. Simul..

[28]  Alexander Kurganov,et al.  Computing MultivaluedSolutions of PressurelessGas Dynamics by Deterministic Particle Methods , 2009 .

[29]  S. Osher,et al.  COMPUTATIONAL HIGH-FREQUENCY WAVE PROPAGATION USING THE LEVEL SET METHOD, WITH APPLICATIONS TO THE SEMI-CLASSICAL LIMIT OF SCHRÖDINGER EQUATIONS∗ , 2003 .

[30]  Nicola Bellomo,et al.  On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives , 2011, SIAM Rev..

[31]  W. Braun,et al.  The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles , 1977 .

[32]  Cécile Appert-Rolland,et al.  Two-way multi-lane traffic model for pedestrians in corridors , 2011, Networks Heterog. Media.

[33]  M. Tidriri,et al.  New Models for the Solution of Intermediate Regimes in Transport Theory and Radiative Transfer: Existence Theory, Positivity, Asymptotic Analysis, and Approximations , 2001 .

[34]  Eitan Tadmor,et al.  A New Model for Self-organized Dynamics and Its Flocking Behavior , 2011, 1102.5575.

[35]  Nicola Bellomo,et al.  Toward a Mathematical Theory of Behavioral-Social Dynamics for Pedestrian Crowds , 2014, 1411.0907.

[36]  Rolf D. Reitz,et al.  One-dimensional compressible gas dynamics calculations using the Boltzmann equation , 1981 .

[37]  Shi Jin,et al.  Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner , 2003 .

[38]  Li Wang,et al.  Contagion Shocks in One Dimension , 2015 .

[39]  Dirk Helbing,et al.  A mathematical model for the behavior of pedestrians , 1991, cond-mat/9805202.

[40]  Feimin Huang,et al.  Well Posedness for Pressureless Flow , 2001 .

[41]  Brett R. Fajen,et al.  Visual navigation and obstacle avoidance using a steering potential function , 2006, Robotics Auton. Syst..

[42]  Stanley Osher,et al.  A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations , 2003 .

[43]  Y. Brenier,et al.  Sticky Particles and Scalar Conservation Laws , 1998 .

[44]  S. M. Deshpande,et al.  Kinetic theory based new upwind methods for inviscid compressible flows , 1986 .

[45]  Christian Dogbé,et al.  Modeling crowd dynamics by the mean-field limit approach , 2010, Math. Comput. Model..

[46]  Milind Tambe,et al.  ESCAPES: evacuation simulation with children, authorities, parents, emotions, and social comparison , 2011, AAMAS.

[47]  Thomas Rey,et al.  An Exact Rescaling Velocity Method for some Kinetic Flocking Models , 2016, SIAM J. Numer. Anal..

[48]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[49]  E. Tadmor,et al.  non-local alignment Critical thresholds in flocking hydrodynamics with , 2014 .

[50]  Hailiang Liu,et al.  Critical Thresholds in a Convolution Model for Nonlinear Conservation Laws , 2001, SIAM J. Math. Anal..

[51]  François Golse,et al.  A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem , 2003 .

[52]  Shi Jin,et al.  On kinetic flux vector splitting schemes for quantum Euler equations , 2011 .

[53]  Debora Amadori,et al.  The one-dimensional Hughes model for pedestrian flow: Riemann—type solutions , 2012 .

[54]  Pierre Degond,et al.  Hybrid kinetic/fluid models for nonequilibrium systems , 2003 .

[55]  Frédéric Coquel,et al.  Well-Posedness and Singular Limit of a Semilinear Hyperbolic Relaxation System with a Two-Scale Discontinuous Relaxation Rate , 2014, Archive for Rational Mechanics and Analysis.

[56]  Shi Jin,et al.  Numerical Approximations of Pressureless and Isothermal Gas Dynamics , 2003, SIAM J. Numer. Anal..