Emerging properties of the degree distribution in large non-growing networks

Jonathan Franceschi, Lorenzo Pareschi, Mattia Zanella (preprint arXiv:2409.06099) 

The degree distribution is a key statistical indicator in network theory, often used to understand how information spreads across connected nodes. In this paper, we focus on non-growing networks formed through a rewiring algorithm and develop kinetic Boltzmann-type models to capture the emergence of degree distributions that characterize both preferential attachment networks and random networks. Under a suitable mean-field scaling, these models reduce to a Fokker-Planck-type partial differential equation with an affine diffusion coefficient, that is consistent with a well-established master equation for discrete rewiring processes.

A Fourth-Order Finite Volume Scheme for Resistive Relativistic Magnetohydrodynamics

Andrea Mignone, Vittoria Berta, Marco Rossazza, Matteo Bugli, Giancarlo Mattia, Luca Del Zanna, Lorenzo Pareschi (Monthly Notices of the Royal Astronomical Society to appear. Preprint arXiv:2407.08519)

We present a finite-volume, genuinely 4th-order accurate numerical method for solving the equations of resistive relativistic magneto-hydrodynamics (Res-RMHD) in Cartesian coordinates. In our formulation, the magnetic field is evolved in time in terms of face-average values via the constrained-transport method while the remaining variables (density, momentum, energy and electric fields) are advanced as cell volume-averages. Spatial accuracy employs 5th-order accurate WENO-Z reconstruction from point values (as described in a companion paper) to obtain left and right states at zone interfaces. Explicit flux evaluation is carried out by solving a Riemann problem at cell interfaces, using the Maxwell-Harten-Lax-van Leer with contact wave resolution (MHLLC).

New trends on the systems approach to modeling SARS-CoV-2 pandemics in a globally connected planet

Giulia Bertaglia, Andrea Bondesan, Diletta Burini, Raluca Eftimie, Lorenzo Pareschi, Giuseppe Toscani (Math. Mod. Meth. App. Sci. to appear. Preprint arXiv:2405.00541)

This paper presents a critical analysis of the literature and perspective research ideas for modeling the epidemics caused by the SARS-CoV-2 virus. It goes beyond deterministic population dynamics to consider several key complexity features of the system under consideration. In particular, the multiscale features of the dynamics from contagion to the subsequent dynamics of competition between the immune system and the proliferating virus. Other topics addressed in this work include the propagation of epidemics in a territory, taking into account local transportation networks, the heterogeneity of the population, and the study of social and economic problems in populations involved in the spread of epidemics.

Instantaneous control strategies for magnetically confined fusion plasma

Giacomo Albi, Giacomo Dimarco, Federica Ferrarese, Lorenzo Pareschi (preprint arXiv:2403.16254)

The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated needs the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this paper, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides a feedback on the equation of motion based on an instantaneous prediction of the discretized system.

Conservative polynomial approximations and applications to Fokker-Planck equations

 

Tino Laidin, Lorenzo Pareschi (preprint arXiv:2402.06473)

We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained polynomial of best approximation for a generic basis of orthogonal polynomials. The construction is entirely general and allows us to derive structure preserving numerical methods for partial differential equations that require the conservation of some moments of the solution, typically representing relevant physical quantities of the problem.