Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions

Evers, Jhm Joep; Hille, Sander C.; Muntean, Adrian

doi:10.1016/j.crma.2013.11.012

Cited by 7 publications

(6 citation statements)

References 9 publications

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“…Previous work in the framework of measures by the authors of the current paper can be found in [11], where apart from the aforementioned force terms also random noise is treated. In [13,14] measure-valued evolutions are treated in the scope of equations of motion that are first-order in time. The link between first-order and second-order models is discussed in [12].…”

Section: Introductionmentioning

confidence: 99%

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From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

et al. 2017

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In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.

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Section: Introductionmentioning

confidence: 99%

“…In [16,17] measure-valued evolutions are treated in the scope of equations of motion that are first-order in time. The link between first-order and second-order models is discussed in [15].…”

Section: Introductionmentioning

confidence: 99%

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

et al. 2017

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“…Recent applications include for instance the porous medium equation [25,32] and the isentropic Euler equations in fluid mechanics [15,32]. Other artcle method are found also in the study of problems related to crowd dynamics and pedestrians flow, see [12,13,26], as well as in the description of the collective motion of large groups of agents, see [4]. Differently from the case of structured population models, the original particle methods are mainly designed for problems where the total mass, or number of individuals, is conserved.…”

Section: Introductionmentioning

confidence: 99%

Optimization in structure population models through the Escalator Boxcar Train

Colombo¹,

Gwiazda²,

Rosińska³

2018

ESAIM: COCV

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The Escalator Boxcar Train (EBT) is a tool widely used in the study of balance laws motivated by structure population dynamics. This paper proves that the approximate solutions defined through the EBT converge to exact solutions. Moreover, this method is rigorously shown to be effective also in computing optimal controls. As preliminary results, the well posedness of classes of PDEs and of ODEs comprising various biological models is also obtained. A specific application to welfare policies illustrates the whole procedure.

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“…Over the last decades they have been successfully applied to solve numerically many problems originated from physics as the Euler equation in fluid mechanics [19,32] and the Vlasov equation in plasma physics [5,12,18]. Recently, the particle methods have been used in problems related to crowd dynamics and flow of pedestrians [17,26,27], models of a collective motion of large groups of agents [8,16,4] and population dynamics [9]. For more applications see [24,25,28,29] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…The scheme under consideration follows a current trend which bases on a kinetic approach to population dynamics problems [2,3,17,21,22,23,26]. Within this approach a population of individuals is divided into groups, which are called cohorts.…”

Section: Introductionmentioning

confidence: 99%

Finite range method of approximation for balance laws in measure spaces

Gwiazda¹,

Orliński²,

Ulikowska³

2017

Kinetic &Amp; Related Models

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In the following paper we reconsider a numerical scheme which has been recently introduced in [10]. The scheme was designed for a wide class of size structured population models with a nonlocal term describing the birth process. We propose a modification of the algorithm, which eliminates the exponential growth in time of the number of particles constituting a numerical solution. Our approach bases on the Finite Range Approximation of the nonlocal term. We provide a convergence theorem, estimates on the convergence speed and results of numerical simulations for several test cases.

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Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions

Cited by 7 publications

References 9 publications

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

Optimization in structure population models through the Escalator Boxcar Train

Finite range method of approximation for balance laws in measure spaces

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