Research

Overview
My research activity focus mainly on phenomena described by time dependent nonlinear partial differential equations, in particular by hyperbolic balance laws and kinetic models. Both classes of equations express fundamental laws of physics and are of great importance in many modern applications that range from gas dynamics, plasma and quantum physics to shallow water, traffic flows, economy and biology, to mention only a few relevant examples. Below are some selected research topics. Other works can be found in the publication page. Slides of some of my recent lectures are also available in the lecture page.

Numerical methods for hyperbolic problems

Hyperbolic systems of balance laws pose several challenging mathematical and numerical problems. Numerical difficulties arise mainly in presence of nonlinear fluxes which may originate shocks and stiff source terms which may pose severe time step restrictions. The crucial point is to derive schemes able to capture the relevant structure of the solutions such as singularities, shocks, instabilities without resolving the small space and time scales.

The research activity concerns different (related) subjects:
  1. Asymptotic preserving schemes for diffusion limits
  2. Implicit-Explicit schemes for time dependent problems
  3. Central schemes for hyperbolic balance laws

    Numerical solution of kinetic equations
    The numerical solution of Boltzmann-type kinetic equations represents a major computational challenge in rarefied gas dynamics and related fields. Typically this is due to the high dimensionality of the problem and to the presence of different time and/or space scales in near-continuum regimes. The necessity of fast solvers for the kinetic integral operators is then an essential part of any numerical schemes for such problems.


    My research interests can be summarized in the following topics:
    1. Spectral and fast methods for kinetic integral operators
    2. Exponential schemes for stiff nonlinear kinetic equations
    3. Monte Carlo methods
    4. Multiscale hybrid methods 

      Kinetic and mean field modelling

      Kinetic equations play a major rule in several applications where the multiscale nature of the phenomena cannot be described by a standard macroscopic approach. They are particularly useful in the study of emergent behaviors in complex systems characterized by the spontaneous formation of spatio-temporal structures as a result of simple local interactions between agents. Complex systems mostly appear in the biological and social contexts but can also be encountered in engineering, physics, chemistry, etc. 

      In the sequel a non-exhaustive list of research fields under study:
      1. Kinetic models of traffic flows 
      2. Mean field modelling in economy and finance 
      3. Modelling emergent behaviors in biology and life sciences