We present a rigorous convergence result for smooth solutions to a singular semilinear hyperbolic approximation, called vector-BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof deeply relies on the dissipative properties of the system and on the use of an energy which is provided by a symmetrizer, whose entries are weighted in a suitable way with respect to the singular perturbation parameter. This strategy allows us to perform uniform energy estimates and to prove the convergence by compactness.
We consider a general model of chemotaxis with finite speed of propagation in one space dimension.
For this model we establish a general result of stability of some constant states both for the Cauchy
problem on the whole real line and for the Neumann problem on a bounded interval.
These results are obtained using the linearized operators and the accurate analysis
of their nonlinear perturbations.
Numerical schemes are proposed to approximate these equations, and the expected
qualitative behavior for large times is compared to several numerical tests.
Mycobacterium tuberculosis (Mtb) is a widely diffused infection. However, in general, the human immune system is able to contain it. In this work, we propose a mathematical model which describes the early immune response to the Mtb infection in the lungs, also including the possible evolution of the infection in the formation of a granuloma. The model is based on coupled reaction-diffusion-transport equations with chemotaxis, which take into account the interactions among bacteria, macrophages and chemoattractant. The novelty of this approach is in the modeling of the velocity field, proportional to the gradient of the pressure developed between the cells, which makes possible to deal with a full multidimensional description and efficient numerical simulations. We perform a linear stability analysis of the model and propose a robust implicit-explicit scheme to deal with long time simulations. Both in one and two-dimensions, we find that there are threshold values in the parameters space, between a contained infection and the uncontrolled bacteria growth, and the generation of granuloma-like patterns can be observed numerically.
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.
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
New computation algorithms for a fluid-dynamic mathematical model
of flows on networks are proposed, described and
First we improve the classical Godunov
scheme (G) for a special flux function,
thus obtaining a more efficient method, the Fast Godunov
scheme (FG) which reduces the number of evaluations for the numerical
Then a new method, namely the Fast Shock Fitting
method (FSF), based on good theorical properties of the solution of the
problem is introduced.
Numerical results and efficiency tests are presented in order to show the
behaviour of FSF in comparison with G, FG and a conservative
scheme of second order.
We consider an integro-differential model for evolutionary game
theory which describes the evolution of a population adopting
mixed strategies. Using a reformulation based on the first
moments of the solution, we prove some analytical properties of
the model and global estimates. The asymptotic behavior and the
stability of solutions in the case of two strategies is analyzed
in details. Numerical schemes for two and three strategies which
are able to capture the correct equilibrium states are also
proposed together with several numerical examples.
We present a one space dimensional model with finite speed of propagation for population dynamics, based both on the
hyperbolic Cattaneo dynamics and the evolutionary game theory. We prove analytical properties of the model and global
estimates for solutions, by using a hyperbolic nonlinear Trotter product formula.