A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis

Adaptive and Recursive Time Relaxed Monte Carlo methods for rarefied gas dynamics

Recently a new class of Monte Carlo methods, called Time Relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equation close to fluid regimes have been introduced. A generalized Wild sum expansion of the solution is at the basis of the simulation schemes. After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium Maxwellian distribution; in this way speed up of the methods close to fluid regimes is obtained by efficiently thermalizing particles close to the equilibrium state. In this work we present an improvement of such methods which allows to obtain an effective uniform accuracy in time without any restriction on the time step and subsequent increase of the computational cost. The main ingredient of the new algorithms is recursivity. Several techniques can be used to truncate the recursive trees generated by the schemes without deteriorating the accuracy of the numerical solution. Techniques based on adaptive strategies are presented. Numerical results emphasize the gain of efficiency of the present simulation schemes with respect to standard DSMC methods

Structure preserving schemes for mean-field equations of collective behavior

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic Problem

Resonate and fire dynamics in Complex Oscillation Based Test of analog filters

Recently, proposals have been made for enhancing the Oscillation Based Test (OBT) methodology by using non-plain oscillation regimes, leading to so called Complex Oscillation Based Test (COBT). Here we focus on a recently illustrated strategy for the testing of analog 2nd order filters, showing that the COBT dynamics is quite similar to that expressed by Resonate & Fire (R+F) neuron models. In this interpretation, the testing approach can be related to firing-rate measures. A brief description is given of the mathematical models necessary to achieve a precise characterization of firing times, showing how it can be used for testing purposes. A practical example with simulation data is also provided. © 2011 IEEE

Solving the Boltzmann equation in N log N

In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension three. These algorithms are implemented for the solution of the Boltzmann equation in two and three dimension, first for homogeneous solutions, then for general non homogeneous solutions. The results are compared to explicit solutions, when available, and to Monte-Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte-Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].Comment: 32 page

Mesoscopic modelling of financial markets

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model (Econ. Lett., 45, (1994), 103--111) using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis

The SXI telescope on board EXIST: scientific performances

The SXI telescope is one of the three instruments on board EXIST, a multiwavelength observatory in charge of performing a global survey of the sky in hard X-rays searching for Supermassive Black Holes. One of the primary objectives of EXIST is also to study with unprecedented sensitivity the most unknown high energy sources in the Universe, like high redshift GRBs, which will be pointed promptly by the Spacecraft by autonomous trigger based on hard X-ray localization on board. The recent addition of a soft X-ray telescope to the EXIST payload complement, with an effective area of ~950 cm2 in the energy band 0.2-3 keV and extended response up to 10 keV will allow to make broadband studies from 0.1 to 600 keV. In particular, investigations of the spectra components and states of AGNs and monitoring of variability of sources, study of the prompt and afterglow emission of GRBs since the early phases, which will help to constrain the emission models and finally, help the identification of sources in the EXIST hard X-ray survey and the characterization of the transient events detected. SXI will also perform surveys: a scanning survey with sky coverage of about 2pi and limiting flux of 5x10^{-14}cgs plus other serendipitous. We give an overview of the SXI scientific performance and also describe the status of its design emphasizing how it has been derived by the scientific requirements.Comment: 9 pages, 6 figures, to be published in Proc. of SPIE, vol 7435-11, 200

First optical validation of a Schwarzschild Couder telescope: the ASTRI SST-2M Cherenkov telescope

The Cherenkov Telescope Array (CTA) represents the most advanced facility designed for Cherenkov Astronomy. ASTRI SST-2M has been developed as a demonstrator for the Small Size Telescope in the context of the upcoming CTA. Its main innovation consists in the optical layout which implements the Schwarzschild-Couder configuration and is fully validated for the first time. The ASTRI SST-2M optical system represents the first qualified example for two mirrors telescope for Cherenkov Astronomy. This configuration permits to (i) maintain a high optical quality across a large FoV (ii) de-magnify the plate scale, (iii) exploit new technological solutions for focal plane sensors. The goal of the paper is to present the optical qualification of the ASTRI SST-2M telescope. The qualification has been obtained measuring the PSF sizes generated in the focal plane at various distance from the optical axis. These values have been compared with the performances expected by design. After an introduction on the Gamma Astronomy from the ground, the optical design and how it has been implemented for ASTRI SST-2M is discussed. Moreover the description of the setup used to qualify the telescope over the full field of view is shown. We report the results of the first--light optical qualification. The required specification of a flat PSF of $\sim 10$ arcmin in a large field of view ~10 deg has been demonstrated. These results validate the design specifications, opening a new scenario for Cherenkov Gamma ray Astronomy and, in particular, for the detection of high energy (5 - 300 TeV) gamma rays and wide-field observations with CTA.Comment: 6 pages, 5 figure