Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations

We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator, the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally we show some numerical results which confirm the theoretical analysis

Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit

We develop a new Monte Carlo method that solves hyperbolic transport equations with stiff terms, characterized by a (small) scaling parameter. In particular, we focus on systems which lead to a reduced problem of parabolic type in the limit when the scaling parameter tends to zero. Classical Monte Carlo methods suffer of severe time step limitations in these situations, due to the fact that the characteristic speeds go to infinity in the diffusion limit. This makes the problem a real challenge, since the scaling parameter may differ by several orders of magnitude in the domain. To circumvent these time step limitations, we construct a new, asymptotic-preserving Monte Carlo method that is stable independently of the scaling parameter and degenerates to a standard probabilistic approach for solving the limiting equation in the diffusion limit. The method uses an implicit time discretization to formulate a modified equation in which the characteristic speeds do not grow indefinitely when the scaling factor tends to zero. The resulting modified equation can readily be discretized by a Monte Carlo scheme, in which the particles combine a finite propagation speed with a time-step dependent diffusion term. We show the performance of the method by comparing it with standard (deterministic) approaches in the literature

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

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

Surface smoothness requirements for the mirrors of the IXO X-ray telescope

The International X-ray Observatory (IXO) is a very ambitious mission, aimed at the X-ray observation of the early Universe. This makes IXO extremely demanding in terms of effective area and angular resolution. In particular, the HEW requirement below 10 keV is 5 arcsec Half-Energy Width (HEW). At higher photon energies, the HEW is expected to increase, and the angular resolution to be correspondingly degraded, due to the increasing relevance of the X-ray scattering off the reflecting surfaces. Therefore, the HEW up to 40 keV is required to be better than 30 arcsec, even though the IXO goal is to achieve an angular resolution as close as possible to 5 arcsec also at this energy. To this end, the roughness of the reflecting surfaces has to not exceed a tolerance, expressed in terms of a surface roughness PSD (Power-Spectral-Density). In this work we provide such tolerances by simulating the HEW scattering term for IXO, assuming a specific configuration for the optical module and different hypotheses on the PSD of mirrors

Bragg concentrators for hard (> 10 keV) X-ray astronomy: Status report

The use of focusing telescopes in hard X-ray (E > 10 keV) astronomy will provide better flux sensitivity and imaging perfomances with respect to the directviewing detectors, utilized until now. We present recent results obtained from our Group regarding the possible use of Bragg-diffraction technique to design hard X-ray focusing telescopes

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

A comprehensive database on synoptic and local circulation over Sicily for mesoscale applications

The aim of this work is to present a database of information available on the island of Sicily (Italy) to be used for the evaluation and/or the calibration of the numerical mesoscale meteorological models. The database relates to land type and land cover of the island as well as to meteorological fields collected at various locations over a time span of various years. The analysis of the database provides information on spatiotemporal variability of characteristic meteorological patterns on the mesoscale range over the island. Specific wind and temperature values characteristic of the regional circulation on the island are presented in the database and analyzed in this paper. The available data have different sources (fixed monitoring stations or measuring campaigns, remote-sensing instruments), and refer to vertical soundings or to measurements at fixed heights. Air temperature, wind speed and wind direction are among the meteorological parameters taken into account. The temporal frequency of the data ranges from 10 minutes to 24 hours