6 research outputs found
Numerical methods for one-dimensional aggregation equations
We focus in this work on the numerical discretization of the one dimensional
aggregation equation \pa_t\rho + \pa_x (v\rho)=0, , in the
attractive case. Finite time blow up of smooth initial data occurs for
potential having a Lipschitz singularity at the origin. A numerical
discretization is proposed for which the convergence towards duality solutions
of the aggregation equation is proved. It relies on a careful choice of the
discretized macroscopic velocity in order to give a sense to the product . Moreover, using the same idea, we propose an asymptotic preserving
scheme for a kinetic system in hyperbolic scaling converging towards the
aggregation equation in hydrodynamical limit. Finally numerical simulations are
provided to illustrate the results
Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity
International audienceWe consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function. 1. Introduction. Macroscopic traffic flow models provide nowadays a validated and powerful approach to simulate and manage traffic on road networks (we refer the interested reader to [25] for an overview of modelling approaches and practical applications and to [11] for a detailed review of the mathematical theory of macroscopic models on networks). These models are based on hyperbolic equations derived from fluid dynamics, and describe the spatio-temporal evolution of macroscopic quantities like vehicle density and mean velocity. One of the seminal macroscopic models was introduced in the mid 1950s by Lighthill and Whitham [21] and Richards [23], who proposed to complete the one-dimensional mass conservation equation ∂ t ρ(t, x) + ∂ x f (t, x) = 0 with a closure relation between speed and density, leading to the fundamental diagram f (t, x) = f (ρ(t, x)) = ρ(t, x)v(ρ(t, x)), which can be derived from empirical speed-density or flow-density data. The classical LWR model therefore reads ∂ t ρ(t, x) + ∂ x (ρ(t, x)v(ρ(t, x))) = 0, x ∈ R, t > 0
Nonlocal Models in Biology and Life Sciences: Sources, Developments, and Applications
Nonlocality is important in realistic mathematical models of physical and
biological systems at small-length scales. It characterizes the properties of
two individuals located in different locations. This review illustrates
different nonlocal mathematical models applied to biology and life sciences.
The major focus has been given to sources, developments, and applications of
such models. Among other things, a systematic discussion has been provided for
the conditions of pattern formations in biological systems of population
dynamics. Special attention has also been given to nonlocal interactions on
networks, network coupling and integration, including models for brain dynamics
that provide us with an important tool to better understand neurodegenerative
diseases. In addition, we have discussed nonlocal modelling approaches for
cancer stem cells and tumor cells that are widely applied in the cell migration
processes, growth, and avascular tumors in any organ. Furthermore, the
discussed nonlocal continuum models can go sufficiently smaller scales applied
to nanotechnology to build biosensors to sense biomaterial and its
concentration. Piezoelectric and other smart materials are among them, and
these devices are becoming increasingly important in the digital and physical
world that is intrinsically interconnected with biological systems.
Additionally, we have reviewed a nonlocal theory of peridynamics, which deals
with continuous and discrete media and applies to model the relationship
between fracture and healing in cortical bone, tissue growth and shrinkage, and
other areas increasingly important in biomedical and bioengineering
applications. Finally, we provided a comprehensive summary of emerging trends
and highlighted future directions in this rapidly expanding field.Comment: 71 page
Differential Models, Numerical Simulations and Applications
This Special Issue includes 12 high-quality articles containing original research findings in the fields of differential and integro-differential models, numerical methods and efficient algorithms for parameter estimation in inverse problems, with applications to biology, biomedicine, land degradation, traffic flows problems, and manufacturing systems