7,433 research outputs found
Stability estimates for scalar conservation laws with moving flux constraints
International audienceWe study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed
Some mathematical problems in numerical relativity
The main goal of numerical relativity is the long time simulation of highly
nonlinear spacetimes that cannot be treated by perturbation theory. This
involves analytic, computational and physical issues. At present, the major
impasses to achieving global simulations of physical usefulness are of an
analytic/computational nature. We present here some examples of how analytic
insight can lend useful guidance for the improvement of numerical approaches.Comment: 17 pages, 12 graphs (eps format
A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation
The paper recalls two of the regularity results for Burgers' equation, and
discusses what happens in the case of genuinely nonlinear, strictly hyperbolic
systems of conservation laws. The first regularity result which is considered
is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the
x-derivative of u when u is an entropy solution of the Cauchy problem for
Burgers' equation with bounded initial data. Its extensions to the case of
systems is then mentioned. The second regularity result of debate is
Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and
generic, in a Baire category sense, initial data are piecewise smooth. The
failure of the same regularity for general genuinely nonlinear systems is next
described. The main focus of this paper is indeed including heuristically an
original counterexample where a kind of stability of a shock pattern made by
infinitely many shocks shows up, referring to [Caravenna-Spinolo] for the
rigorous result.Comment: 10 pages, 1 figur
Modeling the Black Hole Excision Problem
We analyze the excision strategy for simulating black holes. The problem is
modeled by the propagation of quasi-linear waves in a 1-dimensional spatial
region with timelike outer boundary, spacelike inner boundary and a horizon in
between. Proofs of well-posed evolution and boundary algorithms for a second
differential order treatment of the system are given for the separate pieces
underlying the finite difference problem. These are implemented in a numerical
code which gives accurate long term simulations of the quasi-linear excision
problem. Excitation of long wavelength exponential modes, which are latent in
the problem, are suppressed using conservation laws for the discretized system.
The techniques are designed to apply directly to recent codes for the Einstein
equations based upon the harmonic formulation.Comment: 21 pages, 14 postscript figures, minor contents updat
Harmonic Initial-Boundary Evolution in General Relativity
Computational techniques which establish the stability of an
evolution-boundary algorithm for a model wave equation with shift are
incorporated into a well-posed version of the initial-boundary value problem
for gravitational theory in harmonic coordinates. The resulting algorithm is
implemented as a 3-dimensional numerical code which we demonstrate to provide
stable, convergent Cauchy evolution in gauge wave and shifted gauge wave
testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld
boundary conditions and for boundary conditions which explicitly incorporate
constraint preservation. The results are used to assess strategies for
obtaining physically realistic boundary data by means of Cauchy-characteristic
matching.Comment: 31 pages, 14 figures, submitted to Physical Review
Control Problems for Conservation Laws with Traffic Applications
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks
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