Kinetic approximation of a boundary value problem for conservation laws

Aregba-Driollet, Denise; Milisic, Vuk

doi:10.1007/s00211-003-0514-5

Cited by 18 publications

(18 citation statements)

References 46 publications

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“…Following [3,1], we discretize the problem (4.10)-(4.11)-(4.12) and making tend to zero, we obtain a numerical scheme for the initial boundary value problem for the conservation law (4.7), see [1] for more details and convergence results. As usual, we discretize data of the problem by a piecewise constant approximation and we take for k = 1, 2, 3:…”

Section: Numerical Schemementioning

confidence: 99%

Numerical approximations of a traffic flow model on networks

Bretti¹,

Natalini²,

Piccoli³

2006

Networks &Amp; Heterogeneous Media

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We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occurr and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

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Section: Numerical Schemementioning

confidence: 99%

Numerical approximations of a traffic flow model on networks

Bretti¹,

Natalini²,

Piccoli³

2006

Networks &Amp; Heterogeneous Media

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“…For general conservation laws, S. Jin and Z. Xin introduced a relaxation approximation and constructed related numerical schemes, which are equivalent to kinetic schemes with discrete velocities, for the Cauchy problem [17]. A quite complete investigation on second order relaxation and discrete kinetic schemes for general systems of conservation laws in several space variables and with boundary conditions was developed in [1] and [2]. The interactions at junctions are solved by the use of a Linear Programming algorithm that computes the maximized fluxes for all the schemes.…”

Section: Numerical Approximationmentioning

confidence: 99%

“…Concerning the discrete kinetic scheme, we recall that is a quite recent scheme for conservation laws [1,21], applied to traffic flow problem in [7]. The kinetic scheme we consider are known for the Cauchy problem.…”

Section: Numerical Approximationmentioning

confidence: 99%

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A Fluid-Dynamic Traffic Model on Road Networks

Bretti

Natalini

Piccoli

2007

Arch Computat Methods Eng

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We consider a mathematical model for fluiddynamic flows on networks which is based on conservation laws. Road networks are studied as graphs composed by arcs that meet at some nodes, corresponding to junctions, which play a key-role. Indeed interactions occur at junctions and there the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which processes each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

show abstract

“…Following [1] we obtain a numerical scheme for the initial boundary value problem (4)- (6). As usual, we discretize data of the problem by a piecewise constant approximation and we take for k = 1, 2, 3…”

Section: Numerical Schemementioning

confidence: 99%