Article
Mathematics, Applied
A. J. Kriel
Summary: This study introduces a general condition for numerical schemes to mimic the properties of exact solutions of scalar conservation laws. By applying this condition to various schemes, different CFL-like conditions are derived to ensure the accuracy and reliability of the numerical simulations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Thierry Gallay, Arnd Scheel
Summary: We study the long-time behavior of scalar viscous conservation laws by examining the structure of omega-limit sets. We prove that omega-limit sets always consist of constants or shocks, and establish convergence to shocks for arbitrary monotone initial data. In the specific case of Burgers' equation, we review and refine results that parametrize entire solutions using probability measures, and construct initial data where the omega-limit set is not reduced to the translates of a single shock. Finally, we propose several open problems related to describing long-time dynamics.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2023)
Article
Mathematics
Nan Jiang
Summary: In this study, an extension of Yang's convergence criterion is used to demonstrate the entropy convergence of a class of fully discrete a schemes, with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The entropy convergence of the homogeneous counterparts of these schemes has been previously studied, and the author has addressed the convergence when m=2. For semi-discrete a schemes, with or without source terms, the entropy convergence has also been established by the author for m=2.
Article
Mathematics, Applied
Moon-Jin Kang
Summary: This study focuses on the L-2-type contraction property of large perturbations around shock waves of scalar viscous conservation laws with strictly convex fluxes. The contraction is measured by a weighted relative entropy, using an appropriate entropy associated with the strictly convex flux. The results improve upon a recent article on the L-2-contraction property of shocks to scalar viscous conservation laws with a special flux.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2021)
Article
Engineering, Civil
L. Yang, H. Liang, J. Du, S. C. Wong
Summary: This study summarizes the first-order and high-order macroscopic models for solving the pedestrian flow problem and proposes a solution algorithm. The models are rewritten in the form of unified scalar or system hyperbolic conservation laws, and high-order discontinuous Galerkin methods and a second-order fast-sweeping scheme are applied for solving them. Numerical results demonstrate the accuracy and effectiveness of the proposed method and highlight the advantages of triangular meshes.
JOURNAL OF ADVANCED TRANSPORTATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Michael Herty, Niklas Kolbe, Siegfried Mueller
Summary: We propose a novel scheme to numerically solve scalar conservation laws on networks without solving Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure. Appl. Math. 48 (1995), 235-276] and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative and yields well-defined and easy-to-compute coupling conditions even for general networks. We discuss higher order extension of the scheme and applications to traffic flow and two-phase flow. In the former, we compare with results obtained in literature.
NETWORKS AND HETEROGENEOUS MEDIA
(2023)
Article
Mathematics, Applied
Benoit Perthame, Nicolas Seguin, Magali Tournus
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2015)
Article
Engineering, Multidisciplinary
Paola Goatin, Simone Goettlich, Oliver Kolb
ENGINEERING OPTIMIZATION
(2016)
Article
Mathematics, Applied
Sebastien Blandin, Paola Goatin
NUMERISCHE MATHEMATIK
(2016)
Article
Mathematics, Applied
Aekta Aggarwal, Rinaldo M. Colombo, Paola Goatin
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2015)
Article
Mathematics
Maria Laura Delle Monache, Paola Goatin
BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY
(2016)
Article
Mathematics
Aekta Aggarwal, Paola Goatin
BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY
(2016)
Article
Mathematics, Applied
Nina Aguillon, Frederic Lagoutiere, Nicolas Seguin
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Interdisciplinary Applications
Jean-Franois Babadjian, Clement Mifsud, Nicolas Seguin
NETWORKS AND HETEROGENEOUS MEDIA
(2016)
Article
Mathematics, Interdisciplinary Applications
Paola Goatin, Sheila Scialanga
NETWORKS AND HETEROGENEOUS MEDIA
(2016)
Article
Mathematics, Applied
Helene Mathis, Clement Cances, Edwige Godlewski, Nicolas Seguin
JOURNAL OF SCIENTIFIC COMPUTING
(2015)
Article
Mathematics, Applied
Legesse L. Obsu, Maria Laura Delle Monache, Paola Goatin, Semu M. Kassa
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2015)
Article
Mathematics
Paola Goatin
Summary: This survey provides a comprehensive overview of macroscopic models of traffic flow, highlighting their key characteristics and potential limitations. The presentation includes visual illustrations of the models' features. Additionally, open problems and future research directions are presented to stimulate further exploration.
COMMUNICATIONS IN APPLIED AND INDUSTRIAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Paola Goatin, Francesco Rossi
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2017)
Article
Mathematics, Applied
Clement Cances, Frederic Coquel, Edwige Godlewski, Helene Mathis, Nicolas Seguin
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2016)
Article
Mathematical & Computational Biology
Paola Goatin, Matthias Mimault
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2015)