Topological Conformal Dimension

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of quasisymmetric images of the space. We obtain results concerning the behavior of this quantity under products and unions, and compute it for some classical fractals. The range of possible values of the topological conformal dimension is also considered, and we show that this quantity can be fractional.Comment: 16 pages, revised after referee's reports. To appear in Conformal Geometry and Dynamic

Multi-scale control variate methods for uncertainty quantification in kinetic equations

Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. In this paper we consider the construction of novel multi-scale methods for such problems which, thanks to a control variate approach, are capable to reduce the variance of standard Monte Carlo techniques

Fluid Solver Independent Hybrid Methods for Multiscale Kinetic equations

In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I. Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G. Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems. Here, at variance with respect to the previous methods, we construct a method form-fitting to any type of finite volume or finite difference scheme for the reduced equilibrium system. Thanks to the coupling of Monte Carlo techniques for the solution of the kinetic equations with macroscopic methods for the limiting fluid equations, we show how it is possible to solve multiscale fluid dynamic phenomena faster with respect to traditional deterministic/stochastic methods for the full kinetic equations. In addition, due to the hybrid nature of the schemes, the numerical solution is affected by less fluctuations when compared to standard Monte Carlo schemes. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods in comparison with classical approaches used in the simulation of kinetic equations.Comment: 31 page

Fluid Simulations with Localized Boltzmann Upscaling by Direct Simulation Monte-Carlo

In the present work, we present a novel numerical algorithm to couple the Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann equation with a finite volume like method for the solution of the Euler equations. Recently we presented in [14],[16],[17] different methodologies which permit to solve fluid dynamics problems with localized regions of departure from thermodynamical equilibrium. The methods rely on the introduction of buffer zones which realize a smooth transition between the kinetic and the fluid regions. In this paper we extend the idea of buffer zones and dynamic coupling to the case of the Monte Carlo methods. To facilitate the coupling and avoid the onset of spurious oscillations in the fluid regions which are consequences of the coupling with a stochastic numerical scheme, we use a new technique which permits to reduce the variance of the particle methods [11]. In addition, the use of this method permits to obtain estimations of the breakdowns of the fluid models less affected by fluctuations and consequently to reduce the kinetic regions and optimize the coupling. In the last part of the paper several numerical examples are presented to validate the method and measure its computational performances

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

Effects of Transport Delays of Manual Control System Performance

Throughput or transport delays in manual control systems can cause degraded performance and lead to potentially unstable operation. With the expanding use of digital processors, throughput delays can occur in manual control systems in a variety of ways such as in digital flight control systems in real aircraft, and in equation of motion computers and computer generated images in simulators. Research has shown the degrading effect of throughput delays on subjective opinion and system performance and dynamic response. A generic manual control system model is used to provide a relatively simple analysis of and explanation for the effects of various types of delays. The consequence of throughput delays of some simple system architectures is also discussed

Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation

We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis

Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas

We consider the development of Monte Carlo schemes for molecules with Coulomb interactions. We generalize the classic algorithms of Bird and Nanbu-Babovsky for rarefied gas dynamics to the Coulomb case thanks to the approximation introduced by Bobylev and Nanbu (Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Physical Review E, Vol. 61, 2000). Thus, instead of considering the original Boltzmann collision operator, the schemes are constructed through the use of an approximated Boltzmann operator. With the above choice larger time steps are possible in simulations; moreover the expensive acceptance-rejection procedure for collisions is avoided and every particle collides. Error analysis and comparisons with the original Bobylev-Nanbu (BN) scheme are performed. The numerical results show agreement with the theoretical convergence rate of the approximated Boltzmann operator and the better performance of Bird-type schemes with respect to the original scheme