13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme
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B. Maury
Abstract
In this chapter, we consider the pressureless Euler equations with a congestion constraint. This system still raises many open questions and, aside from its onedimensional version, very little is known concerning its solutions. The strategy that we propose relies on previous works on crowd motion models with congestion in the framework of the Wasserstein space, and on a microscopic granular model with nonelastic collisions.We illustrate the approach by preliminary numerical simulations in the two-dimensional setting.
Abstract
In this chapter, we consider the pressureless Euler equations with a congestion constraint. This system still raises many open questions and, aside from its onedimensional version, very little is known concerning its solutions. The strategy that we propose relies on previous works on crowd motion models with congestion in the framework of the Wasserstein space, and on a microscopic granular model with nonelastic collisions.We illustrate the approach by preliminary numerical simulations in the two-dimensional setting.
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
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Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
