41 research outputs found
A possible counterexample to wellposedness of entropy solutions and to Godunov scheme convergence
A particular case of initial data for the two-dimensional Euler equations is
studied numerically. The results show that the Godunov method does not always
converge to the physical solution, at least not on feasible grids. Moreover,
they suggest that entropy solutions (in the weak entropy inequality sense) are
not well-posed
Recent results on the singular local limit for nonlocal conservation laws
We provide an informal overview of recent developments concerning the
singular local limit of nonlocal conservation laws. In particular, we discuss
some counterexamples to convergence and we highlight the role of numerical
viscosity in the numerical investigation of the nonlocal-to-local limit. We
also state some open questions and describe recent related progress.Comment: Proceeding of the "XVII International Conference on Hyperbolic
Problems: Theory, Numerics, Applications.
On interface transmission conditions for conservation laws with discontinuous flux of general shape
International audienceConservation laws of the form with space-discontinuous flux were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of . The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications
Recent results on the singular local limit for nonlocal conservation laws
We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress
One Dimensional Hyperbolic Conservation Laws: Past and Future
Aim of these notes is provide a brief review of the current well-posedness
theory for hyperbolic systems of conservation laws in one space dimension, also
pointing out open problems and possible research directions. They supplement
the slides of the short course given by the author in Erice, May 2023,
available at: sites.google.com/view/erice23/speakers-and-slides.Comment: 38 pages, 25 figure
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Small Collaboration: Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations (hybrid meeting)
Nonlinear hyperbolic partial differential equations constitute a plethora of models from physics, biology, engineering, etc. In this workshop we cover the range from modeling, mathematical questions of well-posedness, numerical discretization and numerical simulations to compare with the phenomenon from nature that was modeled in the first place. Both kinetic and fluid models were discussed
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes