An ill posed Cauchy problem for a hyperbolic system in two space dimensions
An L¹-stability and uniqueness result for balance laws with multifunctions: a model from the theory of granular media
We study the uniqueness and L¹-stability of the Cauchy problem for a 2 × 2 system coming from the theory of granular media [9,10]. We work in a class of weak entropy solutions. The appearance of a multifunction in a source term, given by the Coulomb-Mohr friction law, requires a modification of definition of the weak entropy solution [5,6].
Analysis of Synchronization in a Neural Population by a Population Density Approach
In this paper we deal with a model describing the evolution in time of the density of a neural population in a state space, where the state is given by Izhikevich’s two - dimensional single neuron model. The main goal is to mathematically describe the occurrence of a significant phenomenon observed in neurons populations, the synchronization. To this end, we are making the transition to phase density population, and use Malkin theorem to calculate...
Analysis tools for finite volume schemes
Apparition éventuelle de singularités dans des problèmes d'évolution non linéaires
Arbitrary Lagrangian Eulerian method for compressible plasma simulations
Asymptotic behavior for the centered-rarefaction appearance problem.
Asymptotic behavior of solutions of a system of viscous conservation laws.
Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms.
Asymptotic time-behavior for weighted scalar conservation laws
Asymptotics for conservation laws involving Lévy diffusion generators
Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their -decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.
Boundary layers in weak solutions of hyperbolic conservation laws. III: Vanishing relaxation limits.
Central schemes and contact discontinuities
Central schemes and contact discontinuities
We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes, proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition,...
Central WENO schemes for hyperbolic systems of conservation laws
Central WENO schemes for hyperbolic systems of conservation laws
We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially Non-Oscillatory (WENO) reconstruction of point-values from cell-averages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of Runge-Kutta solvers. We explicitly construct the third and fourth-order scheme and demonstrate...
Central-upwind schemes for the Saint-Venant system
We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes...
Central-Upwind Schemes for the Saint-Venant System
We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central...
Classical solutions to the scalar conservation law with discontinuous initial data
Sufficient and necessary conditions for the existence and uniqueness of classical solutions to the Cauchy problem for the scalar conservation law are found in the class of discontinuous initial data and non-convex flux function. Regularity of rarefaction waves starting from discontinuous initial data and their dependence on the flux function are investigated and illustrated in a few examples.
Compacité par compensation pour une classe de systèmes hyperboliques de p ≥ 3 lois de conservation.
We are concerned with a strictly hyperbolic system of conservation laws ut + f(u)x = 0, where u runs in a region Ω of Rp, such that two of the characteristic fields are genuinely non-linear whereas the other ones are of Blake Temple's type. We begin with the case p = 3 and show, under more or less technical assumptions, that the approximate solutions (uε)ε>0 given either by the vanishing viscosity method or by the Godunov scheme converge to weak entropy solutions as ε goes to 0. The first...