Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations
Roberta Bianchini Roberto Natalini
Kinetic & Related Models 2019, 12(1): 133-158 doi: 10.3934/krm.2019006

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.

keywords: Vector-BGK model discrete velocities incompressible Navier-Stokes equations conservative-dissipative form
Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis
Francesca Romana Guarguaglini Corrado Mascia Roberto Natalini Magali Ribot
Discrete & Continuous Dynamical Systems - B 2009, 12(1): 39-76 doi: 10.3934/dcdsb.2009.12.39
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.
keywords: numerical simulations finite speed of propagation hyperbolic-parabolic systems global stability Chemotactic movements dissipativity finite differences. asymptotic behavior
A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions
Fabrizio Clarelli Roberto Natalini
Mathematical Biosciences & Engineering 2010, 7(2): 277-300 doi: 10.3934/mbe.2010.7.277
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.
keywords: Internal velocity. Reaction-diffusion-advection Multi-dimensional model Mycobacterium tuberculosis Chemotaxis
Conservation laws with discontinuous flux
Mauro Garavello Roberto Natalini Benedetto Piccoli Andrea Terracina
Networks & Heterogeneous Media 2007, 2(1): 159-179 doi: 10.3934/nhm.2007.2.159
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.
keywords: discontinuous flux traffic flow Conservation laws Riemann Solvers front tracking
A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results
Elena Bonetti Cecilia Cavaterra Francesco Freddi Maurizio Grasselli Roberto Natalini
Communications on Pure & Applied Analysis 2019, 18(2): 977-998 doi: 10.3934/cpaa.2019048

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.

keywords: Global existence and uniqueness bulk-surface PDE system sulphation phenomena numerical simulation
Numerical approximations of a traffic flow model on networks
Gabriella Bretti Roberto Natalini Benedetto Piccoli
Networks & Heterogeneous Media 2006, 1(1): 57-84 doi: 10.3934/nhm.2006.1.57
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.
keywords: finite difference schemes boundary conditions. traffic flow fluid-dynamic models Scalar conservation laws
Fast algorithms for the approximation of a traffic flow model on networks
Gabriella Bretti Roberto Natalini Benedetto Piccoli
Discrete & Continuous Dynamical Systems - B 2006, 6(3): 427-448 doi: 10.3934/dcdsb.2006.6.427
New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. 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 flux. 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.
keywords: fluid-dynamic models Scalar conservation laws traffic flow finite difference schemes.
On a continuous mixed strategies model for evolutionary game theory
Astridh Boccabella Roberto Natalini Lorenzo Pareschi
Kinetic & Related Models 2011, 4(1): 187-213 doi: 10.3934/krm.2011.4.187
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.
keywords: Replicator dynamics Evolutionary Game Theory Kinetic equations Continuous mixed strategies Numerical methods.
A hyperbolic model of spatial evolutionary game theory
Anna Lisa Amadori Astridh Boccabella Roberto Natalini
Communications on Pure & Applied Analysis 2012, 11(3): 981-1002 doi: 10.3934/cpaa.2012.11.981
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.
keywords: Prisoner's dilemma. hyperbolic equations Evolutionary game theory Cattaneo model

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