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Nonlinear Fokker--Planck equations play a ma jor role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker--Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin pro jection in the parameter space of the unknowns of the underlying Fokker--Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones. [ABSTRACT FROM AUTHOR]

We consider the development of high-order space and time numerical methods based on implicit-explicit 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. [ABSTRACT FROM AUTHOR]

The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable of considerably accelerating the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates. We show that the additional degrees of freedom can be used to further improve the variance reduction properties of multiscale control variate methods. [ABSTRACT FROM AUTHOR]

This paper deals with the discretization of the compressible Euler system for all Mach number regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency, and they suffer severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic wave speed. In this work, we propose and analyze a new uniformly stable and consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. We propose two space discretizations valid for all regimes. We present several one and two dimensional simulations which show the AP behavior of our schemes, and we observe their L∞ and/or L² stability in all regimes. [ABSTRACT FROM AUTHOR]

We consider the development of high order asymptotic preserving implicit-explicit (IMEX) linear multistep methods for kinetic equations and related problems. The methods are first developed for Bhatnagar-Gross-Krook-like kinetic models and then extended to the case of the full Boltzmann equation. The behavior of the schemes in the Navier-Stokes regime is also studied and compatibility conditions derived. We show that, with respect to IMEX Runge-Kutta methods, the IMEX multistep schemes have several advantages due to the less severe coupling conditions and to the greater computational efficiency. The latter is of paramount importance when dealing with the time discretization of multidimensional kinetic equations. [ABSTRACT FROM AUTHOR]

This paper deals with the development and the analysis of asymptotically stable and consistent schemes in the joint quasi-neutral and fluid limits for the collisional Vlasov-Poisson system. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period and Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scale dynamics and with comparable computational cost with respect to standard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this paper, we propose a first order in time scheme, and we perform a relative linear stability analysis to deal with such problems. The framework we propose will permit us to extend this approach to high order schemes in the near future. Finally, we show the capability of the method in dealing with small scales through numerical experiments [ABSTRACT FROM AUTHOR]

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of BGK type. For Boltzmann-type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques. [ABSTRACT FROM AUTHOR]

In this work we consider the development of a new family of hybrid numerical methods for the solution of kinetic equations which involves different scales. The basic idea is to couple macroscopic and microscopic models in all cases in which the macroscopic model does not provide correct results. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution and a merging of Monte Carlo methods in nonequilibrium regimes with deterministic methods in equilibrium ones. This approach permits us to treat efficiently both the microscopic and the macroscopic scales. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods. [ABSTRACT FROM AUTHOR]