416 research outputs found
Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling
In this work we present a nonlocal conservation law with a velocity depending
on an integral term over a part of the space. The model class covers already
existing models in literature, but it is also able to describe new dynamics
mainly arising in the context of traffic flow modelling. We prove the existence
and uniqueness of weak solutions of the nonlocal conservation law. Further, we
provide a suitable numerical discretization and present numerical examples
Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws
We present a second-order accurate numerical method for a class of nonlocal
nonlinear conservation laws called the "nonlocal pair-interaction model" which
was recently introduced by Du, Huang, and LeFloch. Our numerical method uses
second-order accurate reconstruction-based schemes for local conservation laws
in conjunction with appropriate numerical integration. We show that the
resulting method is total variation diminishing (TVD) and converges towards a
weak solution. In fact, in contrast to local conservation laws, our
second-order reconstruction-based method converges towards the unique entropy
solution provided that the nonlocal interaction kernel satisfies a certain
growth condition near zero. Furthermore, as the nonlocal horizon parameter in
our method approaches zero we recover a well-known second-order method for
local conservation laws. In addition, we answer several questions from the
paper from Du, Huang, and LeFloch concerning regularity of solutions. In
particular, we prove that any discontinuity present in a weak solution must be
stationary and that, if the interaction kernel satisfies a certain growth
condition, then weak solutions are unique. We present a series of numerical
experiments in which we investigate the accuracy of our second-order scheme,
demonstrate shock formation in the nonlocal pair-interaction model, and examine
how the regularity of the solution depends on the choice of flux function.Comment: 23 pages, 15 figures, 5 table
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schr\"odinger Equations
We consider the existence and stability of real-valued, spatially
antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger
equations with fractional dispersion and power-law nonlinearity. As a key
technical result, we demonstrate that the associated linearized operator is
nondegenerate when restricted to antiperiodic perturbations, i.e. that its
kernel is generated by the translational and gauge symmetries of the governing
evolution equation. In the process, we provide a characterization of the
antiperiodic ground state eigenfunctions for linear fractional Schr\"odinger
operators on with real-valued, periodic potentials as well as a
Sturm-Liouville type oscillation theory for the higher antiperiodic
eigenfunctions.Comment: 46 pages, 2 figure
Stability of a Nonlocal Traffic Flow Model for Connected Vehicles
The emerging connected and automated vehicle technologies allow vehicles to
perceive and process traffic information in a wide spatial range. Modeling
nonlocal interactions between connected vehicles and analyzing their impact on
traffic flows become important research questions to traffic planners. This
paper considers a particular nonlocal LWR model that has been studied in the
literature. The model assumes that vehicle velocities are controlled by the
traffic density distribution in a nonlocal spatial neighborhood. By conducting
stability analysis of the model, we obtain that, under suitable assumptions on
how the nonlocal information is utilized, the nonlocal traffic flow is stable
around the uniform equilibrium flow and all traffic waves dissipate
exponentially. Meanwhile, improper use of the nonlocal information in the
vehicle velocity selection could result in persistent traffic waves. Such
results can shed light to the future design of driving algorithms for connected
and automated vehicles.Comment: 20 pages, 5 figure
Lyapunov stabilization for nonlocal traffic flow models
Using a nonlocal second-order traffic flow model we present an approach to
control the dynamics towards a steady state. The system is controlled by the
leading vehicle driving at a prescribed velocity and also determines the steady
state. Thereby, we consider both, the microscopic and macroscopic scales. We
show that the fixed point of the microscopic traffic flow model is
asymptotically stable for any kernel function. Then, we present Lyapunov
functions for both, the microscopic and macroscopic scale, and compute the
explicit rates at which the vehicles influenced by the nonlocal term tend
towards the stationary solution. We obtain the stabilization effect for a
constant kernel function and arbitrary initial data or concave kernels and
monotone initial data. Numerical examples demonstrate the theoretical results
Residual equilibrium schemes for time dependent partial differential equations
Many applications involve partial differential equations which admits
nontrivial steady state solutions. The design of schemes which are able to
describe correctly these equilibrium states may be challenging for numerical
methods, in particular for high order ones. In this paper, inspired by
micro-macro decomposition methods for kinetic equations, we present a class of
schemes which are capable to preserve the steady state solution and achieve
high order accuracy for a class of time dependent partial differential
equations including nonlinear diffusion equations and kinetic equations.
Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure
On a Class of Nonlocal Continuity Equations on Graphs
Motivated by applications in data science, we study partial differential
equations on graphs. By a classical fixed-point argument, we show existence and
uniqueness of solutions to a class of nonlocal continuity equations on graphs.
We consider general interpolation functions, which give rise to a variety of
different dynamics, e.g., the nonlocal interaction dynamics coming from a
solution-dependent velocity field. Our analysis reveals structural differences
with the more standard Euclidean space, as some analogous properties rely on
the interpolation chosen
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