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On the singular local limit for conservation laws with nonlocal fluxes
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In this article, the authors give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19-37), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel.Abstract:
We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.read more
Citations
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Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel
TL;DR: It is proved the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling with respect to the initial data through the doubling of variable technique.
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On Traffic Flow with Nonlocal Flux: A Relaxation Representation
Alberto Bressan,Wen Shen +1 more
TL;DR: In this paper, the authors considered a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density ahead, and obtained uniform BV bounds on the solution.
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On approximation of local conservation laws by nonlocal conservation laws
Alexander Keimer,Lukas Pflug +1 more
TL;DR: In this paper, it was shown that for monotone initial datum the solution of nonlocal conservation laws converges to the entropy solution of the corresponding local conservation laws when the nonlocal reach tends to zero.
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Local limit of nonlocal traffic models: convergence results and total variation blow-up
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On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws
TL;DR: In this article, the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws is investigated, and it is shown that the numerical visco-sensitivity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results.
References
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TL;DR: The CLAWPACK software as discussed by the authors is a popular tool for solving high-resolution hyperbolic problems with conservation laws and conservation laws of nonlinear scalar scalar conservation laws.
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Hyberbolic Conservation Laws in Continuum Physics
TL;DR: In this paper, the authors present a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws, with a focus on balance laws with dissipative source, modeling relaxation phenomena.
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TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
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First order quasilinear equations in several independent variables
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