scholarly journals Crowd motion and evolution PDEs under density constraints

2018 ◽  
Vol 64 ◽  
pp. 137-157 ◽  
Author(s):  
Filippo Santambrogio
Keyword(s):  
Numerical Methods ◽  
Optimal Transport ◽  
Time Discretization ◽  
Existence Results ◽  
Variational Techniques ◽  
Microscopic Models ◽  
Discretization Schemes ◽  
Uniqueness Question ◽  
Crowd Motion

This is a survey about the theory of density-constrained evolutions in theWasserstein space developed by B. Maury, the author, and their collaborators as a model for crowd motion. Connections with microscopic models and other PDEs are presented, as well as several time-discretization schemes based on variational techniques, together with the main theorems guaranteeing their convergence as a tool to prove existence results. Then, a section is devoted to the uniqueness question, and a last one to different numerical methods inspired by optimal transport.

From metastable to coherent sets— Time-discretization schemes

2019 ◽  
Vol 29 (1) ◽  
pp. 012101
Author(s):  
Konstantin Fackeldey ◽  
Péter Koltai ◽  
Peter Névir ◽  
Henning Rust ◽  
Axel Schild ◽  
...  
Keyword(s):  
Time Discretization ◽  
Discretization Schemes

TWO DISCRETIZATION SCHEMES FOR A TIME-DOMAIN DISSIPATIVE ACOUSTICS PROBLEM

2006 ◽  
Vol 16 (10) ◽  
pp. 1559-1598 ◽  
Author(s):  
ALFREDO BERMÚDEZ ◽  
RODOLFO RODRÍGUEZ ◽  
DUARTE SANTAMARINA
Keyword(s):  
Time Domain ◽  
Existence And Uniqueness ◽  
Energy Equation ◽  
Time Discretization ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Fields ◽  
Displacement Fields ◽  
Discretization Schemes ◽  
Initial Boundary Value

This paper deals with a time-domain mathematical model for dissipative acoustics and is organized as follows. First, the equations of this model are written in terms of displacement and temperature fields and an energy equation is obtained. The resulting initial-boundary value problem is written in a functional framework allowing us to prove the existence and uniqueness of solution. Next, two different time-discretization schemes are proposed, and stability and error estimates are proved for both. Finally, numerical results are reported which were obtained by combining these time-schemes with Lagrangian and Raviart–Thomas finite elements for temperature and displacement fields, respectively.

A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs

1998 ◽  
Vol 147 (2) ◽  
pp. 362-387 ◽  
Author(s):  
Gregory Beylkin ◽  
James M. Keiser ◽  
Lev Vozovoi
Keyword(s):  
Time Discretization ◽  
Nonlinear Pdes ◽  
New Class ◽  
Discretization Schemes

scholarly journals On the Accuracy and Efficiency of Transient Spectral Element Models for Seismic Wave Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Sanna Mönkölä
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Seismic Waves ◽  
Time Discretization ◽  
Spectral Element ◽  
Higher Order ◽  
Time Stepping ◽  
Discretization Schemes ◽  
Elastic Wave Equations ◽  
Wave Problems

This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover the coupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on the accuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performed by the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-order time discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrate its function by a numerical simulation.

Nonlinear Aeroelasticity Computations in Transonic Flows Using Tightly Coupling Algorithms

2006 ◽  
Author(s):  
Zhengkun Feng ◽  
Azzeddine Soulai¨mani
Keyword(s):  
Euler Equations ◽  
Compressible Flows ◽  
Schur Complement ◽  
Time Discretization ◽  
Nonlinear Aeroelasticity ◽  
Aeroelastic System ◽  
Discretization Schemes ◽  
Coupling Algorithms ◽  
Coupling Algorithm ◽  
Geometric Conservation Law

A nonlinear computational aeroelasticity model based on the Euler equations of compressible flows and the linear elastodynamic equations for structures is developed. The Euler equations are solved on dynamic meshes using the ALE kinematic description. Thus, the mesh constitutes another field governed by pseudo-elatodynamic equations. The three fields are discretized using proper finite element formulations which satisfy the geometric conservation law. A matcher module is incorporated for the purpose of pairing the grids on the fluid-structure interface and for transferring the loads and displacements between the fluid and structure solvers. Two solutions strategies (Gauss Seidel and Schur-Complement) for solving the nonlinear aeroelastic system are discussed. Using second order time discretization schemes allows us to use large time steps in the computations. The numerical results on the AGARD 445.6 aeroelastic wing compare well with the experimental ones and show that the Schur-complement coupling algorithm is more robust than the Gauss-Seidel algorithm for relatively large oscillation amplitudes.

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