The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

Marco Di Francesco; Graziano Stivaletta

doi:10.1142/S0219891622500205

The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

Marco Di Francesco, Graziano Stivaletta

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We consider the follow-the-leader particle approximation scheme for a 1d scalar conservation law with non-negative compactly supported L∞ initial datum and with a C1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρn is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f(ρ)/ρ)x ≤ 1/t of said condition allows to select a unique weak solution, despite (f(ρ)/ρ)x ≤ 1/t is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition f'(ρ)x ≤ 1/t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f(ρ) = ρ(A-ργ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik-Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.

Original language	English (US)
Pages (from-to)	775-807
Number of pages	33
Journal	Journal of Hyperbolic Differential Equations
Volume	19
Issue number	4
DOIs	https://doi.org/10.1142/S0219891622500205
State	Published - Jan 5 2023
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2023-03-06
Acknowledgements: The authors are grateful to the anonymous referee for her/his comments to the first version of this paper, which led to an improvement of the results. We thank M. D. Rosini for his useful suggestions on the writing of this paper. Part of this work was carried out during the visit of MDF to King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. MDF is deeply grateful for the warm hospitality by people at KAUST, for the excellent scientific environment, and for the support in the development of this work. MDF acknowledges support from the “InterMaths” project of the DISIM Department at the University of L'Aquila.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Analysis
General Mathematics

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10.1142/S0219891622500205

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title = "The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws",

abstract = "We consider the follow-the-leader particle approximation scheme for a 1d scalar conservation law with non-negative compactly supported L∞ initial datum and with a C1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρn is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f(ρ)/ρ)x ≤ 1/t of said condition allows to select a unique weak solution, despite (f(ρ)/ρ)x ≤ 1/t is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition f'(ρ)x ≤ 1/t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f(ρ) = ρ(A-ργ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik-Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.",

author = "{Di Francesco}, Marco and Graziano Stivaletta",

note = "KAUST Repository Item: Exported on 2023-03-06 Acknowledgements: The authors are grateful to the anonymous referee for her/his comments to the first version of this paper, which led to an improvement of the results. We thank M. D. Rosini for his useful suggestions on the writing of this paper. Part of this work was carried out during the visit of MDF to King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. MDF is deeply grateful for the warm hospitality by people at KAUST, for the excellent scientific environment, and for the support in the development of this work. MDF acknowledges support from the “InterMaths” project of the DISIM Department at the University of L'Aquila. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - We consider the follow-the-leader particle approximation scheme for a 1d scalar conservation law with non-negative compactly supported L∞ initial datum and with a C1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρn is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f(ρ)/ρ)x ≤ 1/t of said condition allows to select a unique weak solution, despite (f(ρ)/ρ)x ≤ 1/t is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition f'(ρ)x ≤ 1/t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f(ρ) = ρ(A-ργ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik-Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.

AB - We consider the follow-the-leader particle approximation scheme for a 1d scalar conservation law with non-negative compactly supported L∞ initial datum and with a C1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρn is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f(ρ)/ρ)x ≤ 1/t of said condition allows to select a unique weak solution, despite (f(ρ)/ρ)x ≤ 1/t is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition f'(ρ)x ≤ 1/t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f(ρ) = ρ(A-ργ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik-Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.

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DO - 10.1142/S0219891622500205

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SN - 0219-8916

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EP - 807

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

IS - 4

ER -

The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

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