Roberto Natalini

Convergence to equilibrium for the relaxation approximations of conservation laws

We study the Cauchy problem for 2 × 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws.

Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation

We investigate the relaxation problem for the hydrodynamic isentropic Euler-Poisson system when the momentum relaxation time tends to zero. Very sharp estimates on the solutions, independent of the relaxation time, are obtained and used to establish compactness.

Large time behavior of the solutions to a hydrodynamic model for semiconductors

We establish the global existence of smooth solutions to the Cauchy problem for the one-dimensional isentropic Euler--Poisson (or hydrodynamic) model for semiconductors for small initial data. In particular we show that, as

Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws

t\to\infty

Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory

, these solutions converge to the stationary solutions of the drift-diffusion equations. The existence and uniqueness of stationary solutions to the drift-diffusion equations are proved without the smallness assumption.

Conservation laws with discontinuous flux

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whithams stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

Numerical approximations of a traffic flow model on networks

Summary.We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.

Conservation laws with vanishing nonlinear diffusion and dispersion

We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.

The p53 protein and its molecular network: modelling a missing link between DNA damage and cell fate

We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occur and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

On a relaxation approximation of the incompressible Navier-Stokes equations

We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50s.