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.

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.

Optimization by linear kinetic equations and mean-field Langevin dynamics

Lorenzo Pareschi (Math. Mod. Meth. App. Scie. to appear. Preprint arXiv:2401.05553)

Probably one of the most striking examples of the close connections between global optimization processes and statistical physics is the simulated annealing method, inspired by the famous Monte Carlo algorithm devised by Metropolis et al. in the middle of the last century. In this paper we show how the tools of linear kinetic theory allow to describe this gradient-free algorithm from the perspective of statistical physics and how convergence to the global minimum can be related to classical entropy inequalities. This analysis highlight the strong link between linear Boltzmann equations and stochastic optimization methods governed by Markov processes.

Reduced variance random batch methods for nonlocal PDEs

Lorenzo Pareschi, Mattia Zanella (to appear in Acta Applicandae Mathematicae. Preprint arXiv:2401.00493)

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy.