Discrete-time Flow Research Articles

Arbitrarily High-Order Conservative Schemes for the Generalized Korteweg--de Vries Equation

This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg--de Vries (KdV) equation. This approach is based on the scalar auxiliary variable method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge--Kutta method will preserve all the three invariants (momentum, mass, and the reformulated energy) in the discrete time flow (assuming the spatial variable is continuous). With the Fourier pseudospectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the space-time full discrete sense. The discrete mass possesses the precision of the spectral accuracy. Our numerical experiment shows the great efficiency of this scheme in simulating the breathers for the modified KdV equation.

Evacuation Contraflow Problems with Not Necessarily Equal Transit Time on Anti-parallel Arcs

An evacuation planning problem provides a plan for existing road topology that sends maximum number of evacuees from risk zone to the safe destination in minimum time period during disasters. The problems with different road network attributes have been studied, and solutions have been proposed in literature. Evacuation planning problems with network contraflow approach, reversing the direction of traffic flow on lanes, with the same transit time on anti-parallel arcs have also been extensively studied. The approach, due to its lane-direction reversal property, can be taken as a potential remedy to mitigate congestion and reduce casualties during emergencies. In this paper, we propose a mathematical optimization contraflow model for the evacuation problem with the case where there may exist different transit time on anti-parallel arcs. We also propose analytical solutions to a few variants of problems, such as maximum dynamic contraflow problem and earliest arrival contraflow problem in which arc reversal capability is allowed only once at time zero. We extend the solution to solve the problems with continuous time settings by applying the natural relation between discrete time flows and continuous time flows. The solution procedures are based on application of temporally repeated flows (TRFs) on modified network, and they solve the problems optimally in strongly polynomial time.

Shrinking target problems for flows on homogeneous spaces

We study shrinking targets problems for discrete time flows on a homogenous space $\Gamma\backslash G$ with $G$ a semisimple group and $\Gamma$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

Shrinking targets for discrete time flows on hyperbolic manifolds

We prove dynamical Borel Canteli Lemmas for discrete time homogenous flows hitting a sequence of shrinking targets in a hyperbolic manifold. These results apply to both diagonalizable and unipotent flows, and to any family of measurable shrinking targets in the manifold. As a special case, we establish logarithm laws for the first hitting times to shrinking balls and shrinking cusp neighborhoods, refining and improving on perviously known results.

A discrete time dynamic flow model and a formulation and solution method for dynamic route choice

This paper considers the ideal dynamic user optimal (DUO) route choice problem using a route-based variational inequality approach. A discrete time dynamic flow model is developed, which uses link travel time functions to determine time-dependent network states. The proposed flow model is built on discrete time flow variables, to eliminate the discretization process of continuous time based models. Continuity of route travel time functions is proven to establish the existence of a solution, on the condition that the link travel time functions are continuous. Furthermore, flow dispersion and concentration can be simulated, which is expected to enhance the ability of capturing dynamics of traffic movements. A variational inequality formulation based on an alternative cost mapping is proposed, which is derived from a route swapping heuristic approach. As a solution method, the projection-based approach is used since the route travel time functions in our model are not smooth. To increase the performance of the projection-based methods, an efficient implementation of the projection operation is developed. Computational experiences with two example networks are provided to illustrate the model.