Implicit-Explicit Methods for Evolutionary Partial Differential Equations

Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo, SIAM Mathematical Modelling and Computations Series, 2024 

Publishing December 2024

Implicit-explicit (IMEX) time discretization methods have proven to be highly effective for the numerical solution of a wide class of evolutionary partial differential equations (PDEs) across various contexts. These methods have become mainstream for solving evolutionary PDEs, particularly in the fields of hyperbolic and kinetic equations. The first book on the subject, Implicit-Explicit Methods for Evolutionary Partial Differential Equations provides an in-depth yet accessible approach. The authors summarize and illustrate the construction, analysis, and application of IMEX methods using examples, test cases, and implementation details; guide readers through the various methods and teach them how to select and use the one most appropriate for their needs; and demonstrate how to identify stiff terms and effectively implement high-order methods in time for a variety of systems of PDEs. Readers interested in learning modern techniques for the effective numerical solution of evolutionary PDEs with multiple time scales will find in this book a unified, compact, and accessible treatment.

Kinetic models for optimization: a unified mathematical framework for metaheuristics

 Giacomo Borghi, Michael Herty, Lorenzo Pareschi (preprint arXiv:2410.10369)

Metaheuristic algorithms, widely used for solving complex non-convex and non-differentiable optimization problems, often lack a solid mathematical foundation. In this review, we explore how concepts and methods from kinetic theory can offer a potential unifying framework for a variety of metaheuristic optimization methods. By applying principles from collisional and non-collisional kinetic theory, we outline how particle-based algorithms like Simulated Annealing, Genetic Algorithms, Particle Swarm Optimization, and Ensemble Kalman Filter may be described through a common statistical perspective. This approach not only provides a path to deeper theoretical insights and connects different methods under suitable asymptotic scalings, but also enables the derivation of novel algorithms using alternative numerical solvers. While not exhaustive, our review highlights how kinetic models can enhance the mathematical understanding of existing optimization algorithms and inspire new computational strategies.

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.