8 research outputs found
Is the Classic Convex Decomposition Optimal for Bound-Preserving Schemes in Multiple Dimensions?
Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120,
2010], the Zhang--Shu framework has attracted extensive attention and motivated
many bound-preserving (BP) high-order discontinuous Galerkin and finite volume
schemes for various hyperbolic equations. A key ingredient in the framework is
the decomposition of the cell averages of the numerical solution into a convex
combination of the solution values at certain quadrature points, which helps to
rewrite high-order schemes as convex combinations of formally first-order
schemes. The classic convex decomposition originally proposed by Zhang and Shu
has been widely used over the past decade. It was verified, only for the 1D
quadratic and cubic polynomial spaces, that the classic decomposition is
optimal in the sense of achieving the mildest BP CFL condition. Yet, it
remained unclear whether the classic decomposition is optimal in multiple
dimensions. In this paper, we find that the classic multidimensional
decomposition based on the tensor product of Gauss--Lobatto and Gauss
quadratures is generally not optimal, and we discover a novel alternative
decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and
3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP
time step-size than the classic one, and moreover, it is rigorously proved to
be optimal to attain the mildest BP CFL condition, yet requires much fewer
nodes. The discovery of such an optimal convex decomposition is highly
nontrivial yet meaningful, as it may lead to an improvement of high-order BP
schemes for a large class of hyperbolic or convection-dominated equations, at
the cost of only a slight and local modification to the implementation code.
Several numerical examples are provided to further validate the advantages of
using our optimal decomposition over the classic one in terms of efficiency
On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws
This paper presents the first systematic study on the fundamental problem of
seeking optimal cell average decomposition (OCAD), which arises from
constructing efficient high-order bound-preserving (BP) numerical methods
within Zhang--Shu framework. Since proposed in 2010, Zhang--Shu framework has
attracted extensive attention and been applied to developing many high-order BP
discontinuous Galerkin and finite volume schemes for various hyperbolic
equations. An essential ingredient in the framework is the decomposition of the
cell averages of the numerical solution into a convex combination of the
solution values at certain quadrature points. The classic CAD originally
proposed by Zhang and Shu has been widely used in the past decade. However, the
feasible CADs are not unique, and different CAD would affect the theoretical BP
CFL condition and thus the computational costs. Zhang and Shu only checked, for
the 1D and spaces, that their classic CAD based on
the Gauss--Lobatto quadrature is optimal in the sense of achieving the mildest
BP CFL conditions.
In this paper, we establish the general theory for studying the OCAD problem
on Cartesian meshes in 1D and 2D. We rigorously prove that the classic CAD is
optimal for general 1D spaces and general 2D spaces
of arbitrary . For the widely used 2D spaces, the classic CAD
is not optimal, and we establish the general approach to find out the genuine
OCAD and propose a more practical quasi-optimal CAD, both of which provide much
milder BP CFL conditions than the classic CAD. As a result, our OCAD and
quasi-optimal CAD notably improve the efficiency of high-order BP schemes for a
large class of hyperbolic or convection-dominated equations, at little cost of
only a slight and local modification to the implementation code
A Diffuse-Domain Based Numerical Method for a Chemotaxis-Fluid Model
In this paper, we consider a coupled chemotaxis-fluid system that models
self-organized collective behavior of oxytactic bacteria in a sessile drop.
This model describes the biological chemotaxis phenomenon in the fluid
environment and couples a convective chemotaxis system for the oxygen-consuming
and oxytactic bacteria with the incompressible Navier-Stokes equations subject
to a gravitational force, which is proportional to the relative surplus of the
cell density compared to the water density.
We develop a new positivity preserving and high-resolution method for the
studied chemotaxis-fluid system. Our method is based on the diffuse-domain
approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD)
model for simulating bioconvection in complex geometries. The drop domain is
imbedded into a larger rectangular domain, and the original boundary is
replaced by a diffuse interface with finite thickness. The original
chemotaxis-fluid system is reformulated on the larger domain with additional
source terms that approximate the boundary conditions on the physical
interface. We show that the cf-DD model converges to the chemotaxis-fluid model
asymptotically as the width of the diffuse interface shrinks to zero. We
numerically solve the resulting cf-DD system by a second-order hybrid
finite-volume finite-difference method and demonstrate the performance of the
proposed approach on a number of numerical experiments that showcase several
interesting chemotactic phenomena in sessile drops of different shapes, where
the bacterial patterns depend on the droplet geometries
Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments
Traffic waves are phenomena that emerge when the vehicular density exceeds a
critical threshold. Considering the presence of increasingly automated vehicles
in the traffic stream, a number of research activities have focused on the
influence of automated vehicles on the bulk traffic flow. In the present
article, we demonstrate experimentally that intelligent control of an
autonomous vehicle is able to dampen stop-and-go waves that can arise even in
the absence of geometric or lane changing triggers. Precisely, our experiments
on a circular track with more than 20 vehicles show that traffic waves emerge
consistently, and that they can be dampened by controlling the velocity of a
single vehicle in the flow. We compare metrics for velocity, braking events,
and fuel economy across experiments. These experimental findings suggest a
paradigm shift in traffic management: flow control will be possible via a few
mobile actuators (less than 5%) long before a majority of vehicles have
autonomous capabilities
Tracking vehicle trajectories and fuel rates in phantom traffic jams: Methodology and data
International audienceHigh-fidelity vehicle trajectory data is becoming increasingly important in traffic modeling, especially to capture dynamic features such as stop-and-go waves. This article presents data collected in a series of eight experiments on a circular track with human drivers. The data contains smooth flowing and stop-and-go traffic conditions. The vehicle trajectories presented in this article are collected using a panoramic 360-degree camera, and fuel rate data is recorded via an on-board diagnostics scanner installed in each vehicle. The video data from the 360-degree camera is processed with an offline unsupervised algorithm to extract vehicle trajectories from experimental data. The trajectories are highly accurate, with a mean positional bias of less than 0.01 m and a standard deviation of 0.11 m. The velocities are also validated to be highly accurate with a bias of 0.02 m/s and standard deviation of 0.09 m/s. The source code and data used in this article are published with this work