Giacomo Dimarco

Numerical methods for kinetic equations

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

Exponential Runge-Kutta Methods for Stiff Kinetic Equations

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.

A multiscale kinetic-fluid solver with dynamic localization of kinetic effects

This paper collects the efforts done in our previous works [7,9,10] to build a robust multiscale kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems which present non-equilibrium localized regions that can move, merge, appear or disappear in time. The main ingredients of the present work are the followings ones: a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary; this multiscale description of the flow is obtained by using a micro-macro decomposition of the distribution function [9]; the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function; to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself. Several numerical examples are presented to validate the method and measure its computational efficiency.

Asymptotic Preserving Implicit-Explicit Runge--Kutta Methods for Nonlinear 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.

Hybrid Multiscale Methods II. Kinetic Equations

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.

A moving interface method for dynamic kinetic-fluid coupling

This paper is a continuation of earlier work [P. Degond, S. Jin, L. Mieussens, A smooth transition between kinetic and hydrodynamic equations, Journal of Computational Physics 209 (2005) 665-694] in which we presented 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. Discontinuities or sharp gradients of the solution are responsible for locally strong departures to local equilibrium which require the resolution of the kinetic model. 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. In the present paper, we specifically consider the possibility of moving the kinetic region or creating new kinetic regions, by evolving the cut-off function with respect to time. We present algorithms which perform this task by taking into account indicators which characterize the non-equilibrium state of the gas. The method is shown to be highly flexible as it relies on the time evolution of the buffer cut-off function rather than on the geometric definition of a moving interface which requires remeshing, by contrast to many previous methods. Numerical examples are presented which validate the method and demonstrate its performances.

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.

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.

An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas

In this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present. The method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones. In this paper we extend Domain Decomposition techniques, obtained through dynamic coupling and buffer zones, to the study of plasmas and, moreover, we combine them with Asymptotic Preserving and Asymptotically Accurate strategies for the time integration. We use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods, in order to guarantee high efficiency and accuracy in all regimes. The numerical scheme is validated and its performances are analyzed by numerical simulations.

Implicit-Explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors

In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diusion equation. Our aim is to develop a method which accurately works for the dierent regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suer from the usual parabolic stiness in the diusive limit. We show dierent numerical results which permit to appreciate the performances of the proposed schemes.