Research Interests:

Development of efficient and accurate Monte Carlo and hybrid methods. (with Prof. Lorenzo Pareschi, University of Ferrara, Italy and Prof. Pierre Degond, University Paul Sabatier, Toulouse France)
Kinetic models involve large difficulties in terms of numerical methods and computations. To address these issues, in real simulations, probabilistic methods such as Direct Simulation Monte Carlo (DSMC) are often used, for their great flexibility, ability in dealing with different collisional terms and low computational cost. On the other hand, solutions provided by these schemes are affected by large fluctuations and, in unsteady situations, the impossibility to average inevitably leads to or inaccurate solutions or very expensive simulations. To this aim our work consists in the development of hybrid and dynamic domain decomposition techniques that, in the recent past, have given very good results. We are extending the methods, already implemented in the context of the BGK model, to the Boltzmann case through the use of time relaxed Monte Carlo. These methods allow to expand, through suitable power series, the distribution function and replace the high order terms with the local equilibrium. The results obtained so far are very promising in terms of computational cost compared to traditional techniques for deterministic kinetic equations as for instance discrete velocity methods or spectral methods. In addition with these methods it is possible to provide solutions containing less fluctuations which completely disappear in the fluid limit. In addition, at the present moment we are working on variance reduction techniques methods for DSMC, trying to drive particles to match macroscopic moments provided by appropriate fluid dynamic solver.

Mathematical Modeling in Rarefied Gas Dynamics (with Prof. Pierre Degond University Paul Sabatier, Toulouse France and Prof. Luc Mieussens, University of Bordeaux France)
We developed an automatic domain decomposition method for the solution of gas dynamics problems which require a localized resolution of the kinetic scale. The basic idea is to couple the macroscopic hydrodynamics model and the microscopic kinetic model through a buffer zone in which both equations are solved. The buffer zone is drawn around the kinetic region by introducing a cut-off function, which takes values between zero and one and which is identically zero in the fluid zone and one in the kinetic zone. We specifically considered the possibility of moving the kinetic region or creating new kinetic regions, by evolving the cut-off function with respect to time. The possibility to obtain narrow kinetic regions inside the domain localized around the non-equilibrium zones which move in time permits strong computational speed-up without losing the accuracy of the solution. This coupling will join both deterministic and stochastic methods for the kinetic equations with high order schemes for the fluid ones. We are currently try to face with the following challenges: the degradation of the fluid solver quality and the larger difficulties in the definition of the different domains due to the coupling with a particle solver.

Modelling and numerical methods for chemically reacting gas mixtures and plasmas. (with Prof. Russel Caflisch and Dott. Richard Wang, University of California Los Angeles, Prof Bruce Cohen and Dott. Andris Dimits of Lawrence Livermore National Laboratory, California and Dott. Marzia Bisi and Prof. Giampiero Spiga, University of Parma, Italy)
We concentrated on the derivation of kinetic models, on the study of suitable hydrodynamic closures, and on the construction of suitable numerical schemes and simulations. We would like to introduce new kinetic models of time-relaxation type which preserves collision invariants of the reactive Boltzmann equations, the mass action law of chemical equilibrium and that, unlike other BGK models available in the literature, may be used also to describe phenomena characterized by "fast" chemical reactions, namely when chemical characteristic time is of the same order of magnitude as the mechanical one. We are investigating the problem of relaxation to the equilibrium state looking for rates of convergence to the equilibrium by "entropy/entropy dissipation" techniques. We are also considering possible hydrodynamic limits as asymptotic limits with respect to small parameters, to different levels of accuracy (Euler, Navier-Stokes, etc.). Applications cover both classical fluid dynamic problems such as the Riemann problem, shock waves, detonations, and both more recent issues as the recombination of ions and electrons in mixtures of plasmas and chemical reactions in nuclear reactor. We also want to extend classical Monte Carlo methods for the Boltzmann equation to the case of plasmas described by Landau-Fokker-Planck equations. We have already obtained some preliminary encouraging results with these methods. It is important to remark that, until now, the majority of codes that describe plasma physics problems do not take into account collisions, while many recent results obtained by physicists state that collisions are an important phenomenon which has to be accounted for in many applications such as nuclear fusion in Tokamaks or ITER.

Development of high order discretizations for multiscale problems. (with Prof. Lorenzo Pareschi, University of Ferrara, Italy)
As an alternative to Monte Carlo methods one can consider deterministic techniques which are particularly interesting for non stationary problems. From a mathematical viewpoint, several of the problems under consideration are described by systems of hyperbolic differential equations with relaxation terms. The development of efficient numerical schemes for these problems is considered a major challenge in numerical analysis, since in many applications the relaxation time varies by several orders of magnitude. In the frequent cases, where it is difficult to divide the problem in regions with different regimes and to use different numerical schemes for the different areas, it is necessary to solve a multiscale relaxation system in the full domain. In these situations, splitting methods are often used because of their simplicity and robustness, but on the other hand, in general, with these strategies it is difficult to obtain high order accuracy. For this reason our aim is to consider Implicit-Explicit (IMEX) Runge-Kutta methods, which allow to obtain high order schemes and which are capable to satisfy some suitable properties such as strong-stability-preserving (SSP), a monotonicity property for the entropy, and asymptotic preserving (AP) for the limit case where the relaxation parameter vanishes. When we try to apply implicit schemes to kinetic operators we often encounter the following difficulties: stiff source term which may originate a dense nonlinear system, fully implicit scheme which gives prohibitive computational costs and, except for first order methods, implicit Runge-Kutta do not satisfy the strong stability preserving (SSP) property which in the kinetic case it turns to be equivalent to the entropy property. To that aim, we want to introduce for the numerical integration in time of the kinetic equations, exponential methods which in general guarantee: positivity, conservations of macroscopic quantities and asymptotic preservation. It appears to be possible, with this methods, to construct high order (grater than two) which also satisfy both AP and SSP properties.

INVOLVEMENTS IN INTERNATIONAL AND NATIONAL RESEARCH PROGRAMS

HYKE 2002-2004: HYperbolic and Kinetic Equations : Asymptotics, Numerics, Analysis . Coordinator Prof. Norbert Mauser.

MURST-PRIN 2005: Advanced Numerical methods for evolution equations. Coordinator Prof. Alfio Quarteroni.

MIUR-Programma Vigoni 2007: Numerical Methods for the simulation and the optimization of traffic flow on road network. Coordinator Prof. Lorenzo Pareschi.

University Italo-Francese Galileo 2006: Air Pollution due to Powder: mathematical problems and numerical simulations. Coordinator Prof. M. Groppi.

MAE - Italy-South Africa Program 2008-2010: Particulate air pollution: numerical methods and simulations. Coordinator Prof. L.Pareschi.

FAR - Ferrara University 2008: Numerical Model and Advanced Simulation Techniques for Partial Di®erential Equations. Coordinator Prof. L.Pareschi.

MURST-PRIN 2007: Advanced Numerical methods for evolution equations and multiscale problems. Coordinator Prof. Alfio Quarteroni.