Numerical simulations of the Euler system with congestion constraint

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [6,17] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [16,18] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes.

[1]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[2]  Parviz Moin,et al.  A semi-implicit method for resolution of acoustic waves in low Mach number flows , 2002 .

[3]  Pierre Degond,et al.  A Model for the Formation and Evolution of Traffic Jams , 2008 .

[4]  Randall J. LeVeque,et al.  THE DYNAMICS OF PRESSURELESS DUST CLOUDS AND DELTA WAVES , 2004 .

[5]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[6]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

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

[8]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[9]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[10]  Chia-Jung Hsu.  Numerical Heat Transfer and Fluid Flow , 1981 .

[11]  Harumi Hattori,et al.  The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion — Isothermal case , 1986 .

[13]  Pierre Degond,et al.  Modelling and simulation of vehicular traffic jam formation , 2008 .

[14]  Michael Breuß,et al.  A relaxation scheme for the approximation of the pressureless Euler equations , 2006 .

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

[16]  P. Degond,et al.  All speed scheme for the low Mach number limit of the isentropic Euler equations , 2009, 0908.1929.

[17]  R. Klein.  Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics , 1995 .

[18]  Pierre Degond,et al.  MACH-NUMBER UNIFORM ASYMPTOTIC-PRESERVING GAUGE SCHEMES FOR COMPRESSIBLE FLOWS , 2007 .

[19]  H. Guillard,et al.  On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes , 2004 .

[20]  Bart Merci,et al.  A conservative discrete compatibility-constraint low-Mach pressure-correction algorithm for time-accurate simulations of variable density flows , 2009, J. Comput. Phys..

[21]  P. Wesseling,et al.  A conservative pressure-correction method for flow at all speeds , 2003 .

[22]  Pierre Degond,et al.  Congestion in a Macroscopic Model of Self-driven Particles Modeling Gregariousness , 2009, 0908.1817.

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

[24]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

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

[26]  Florent Berthelin,et al.  EXISTENCE AND WEAK STABILITY FOR A PRESSURELESS MODEL WITH UNILATERAL CONSTRAINT , 2002 .

[27]  Salissou Moutari,et al.  A traffic flow model with constraints for the modeling of traffic jams , 2008 .

[28]  E. Turkel,et al.  PRECONDITIONING TECHNIQUES IN COMPUTATIONAL FLUID DYNAMICS , 1999 .

[29]  Stanley Osher,et al.  A second order primitive preconditioner for solving all speed multi-phase flows , 2005 .

[30]  Yann Brenier,et al.  A Hierarchy of Models for Two-Phase Flows , 2000, J. Nonlinear Sci..

[31]  Hester Bijl,et al.  A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates , 1998 .

[32]  HAILIANG LIU,et al.  FORMATION OF DELTA-SHOCKS AND VACUUM STATES INTHE VANISHING PRESSURE , 2001 .

[33]  A. A. Amsden,et al.  A numerical fluid dynamics calculation method for all flow speeds , 1971 .

[34]  Xue-song Li,et al.  An All-Speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour , 2008, J. Comput. Phys..

[35]  Alexander Kurganov,et al.  A New Sticky Particle Method for Pressureless Gas Dynamics , 2007, SIAM J. Numer. Anal..

[36]  C. Munz,et al.  The extension of incompressible flow solvers to the weakly compressible regime , 2003 .

[37]  N. Bellomo,et al.  ON THE MODELLING CROWD DYNAMICS FROM SCALING TO HYPERBOLIC MACROSCOPIC MODELS , 2008 .

[38]  E. Dick,et al.  Mach-uniformity through the coupled pressure and temperature correction algorithm , 2005 .

[39]  Pierre-Louis Lions,et al.  On a free boundary barotropic model , 1999 .