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• Uncertainty quantification for the Boltzmann equation. • Variance reduction of standard Monte Carlo sampling methods. • Control variate methods based on reduced models obtained in a multi-scale setting. • Asymptotic behavior of the control variate methods and fluid-dynamic limit. • Error estimates show a strong acceleration of the convergence with respect to standard Monte Carlo. 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. [ABSTRACT FROM AUTHOR]

We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. Curl-grad operators follow as a consequence. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework. Numerical examples show that the theoretical order of convergence is reached by this new class of methods, and prove the exact preservation of the incompressibility constraint. • High order discontinuous Galerkin divergence, curl and gradient operators. • Structure-preserving div-curl operator. • 3D Cartesian meshes. • Implicit-Explicit (IMEX) scheme for vortex-strem incompressible Navier-Stokes. • Compatible numerical dissipation. [ABSTRACT FROM AUTHOR]

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26] ) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the extension of the method developed in Dimarco and Loubère (2013) [22,23] with the aim of facing the numerical resolution of multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere. This is the case of many realistic applications: some regions of the computational domain require a microscopic description, given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type. Our aim is to show how the kinetic scheme developed in the pre-cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi-scale realistic problems. Several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate the capability of the fast semi-Lagrangian scheme developed in [20] and [21] to deal with parallel architectures. First, we will present the behaviors of such scheme on a classical architecture using OpenMP and then on GPU (Graphics Processing Unit) architecture using CUDA. The goal is to prove that this new scheme is well adapted to these types of parallelizations, and, moreover that the gain in CPU time is substantial on nowadays affordable computers. We first present the sequential version of our high-order kinetic scheme and focus on important details for an effective parallel implementation. Then, we introduce the specific treatments and algorithms which have been developed for an OpenMP and CUDA parallelizations. Numerical tests are shown for the full 3D/3D simulations. These assess the important speed-up factor of the method gained between the sequential code and the parallel versions and its very good scalability which makes this approach a real competitor with respect to existing schemes for the solution of multidimensional kinetic models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the Boltzmann type (Dimarco and Loubere, 2013) [17]. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time. On the contrary to classical semi-Lagrangian methods one does not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six-dimensional case without parallelization. However, the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit. In the present work, we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes. In particular, in the fluid limit, the method automatically degenerates into a high order method for the compressible Euler equations. Numerical examples are presented which validate the method, show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well-known schemes (Direct Simulation Monte Carlo, Finite Volume, MUSCL, WENO). [Copyright &y& Elsevier]

Abstract: In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the authorʼs knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D. [Copyright &y& Elsevier]

Abstract: 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 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 . 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. [Copyright &y& Elsevier]

Abstract: This paper collects the efforts done in our previous works 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 ; 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

• High order space and time numerical methods on arbitrary unstructured polygonal meshes in a multidimensional setting. • A CWENO reconstruction reduces stencil size and computational cost. This is of paramount importance for kinetic models. • Implicit-Explicit Runge-Kutta schemes in the finite volume framework are studied. • Asymptotic preservation and accuracy in the fluid limit is shown. • Numerical results confirm the excellent behavior of the method opening the path to simulation of realistic physical problems. In this work, a family of high order accurate Central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear kinetic equation of relaxation type is presented. After discretization of the velocity space by using a discrete ordinate approach, the space reconstruction is realized by integration over conformal arbitrary shaped closed space control volumes in a CWENO fashion. Compared to other WENO methods on unstructured meshes, in the method here presented, the total stencil size is the minimum possible and the linear weights can be arbitrarily chosen. These two aspects make their use for kinetic equations and the practical implementation on general unstructured meshes particularly interesting. The full discretization is then obtained by combining the previous phase-space approximation with an Implicit-Explicit Runge Kutta high order time discretization which guarantees stability, accuracy and preservation of the asymptotic state. In particular, to guarantee in the finite volume framework space accuracy higher than two, a new class of IMEX methods has been set into place and its properties have been studied. The formal order of accuracy is numerically measured for different regimes, computational performances of the proposed class of methods are tested on several standard two dimensional benchmark problems for kinetic equations. The novel methods are finally applied to a prototype engineering problem consisting in a supersonic flow around a NACA 0012 airfoil. In our computations we employ up to ≈325 millions of degrees of freedom and 256 GB of RAM run on 128 cores with Fortran-MPI providing evidence that the above schemes are suitable for implementation on parallel distributed memory supercomputers. [ABSTRACT FROM AUTHOR]

• All Mach solver for the Euler equations. • Asymptotic Preserving discretization. • Fully second order space-time method. • Collocated numerical scheme on a 3D Cartesian grid. This article deals with the development of a numerical method for the compressible Euler system valid for all Mach numbers: from extremely low to high regimes. In classical fluid dynamic problems, one faces both situations in which the flow is subsonic, and consequently acoustic waves are very fast compared to the velocity of the fluid, and situations in which the fluid moves at high speed and compressibility may generate shock waves. Standard explicit fluid solvers such as Godunov method fail in the description of both flows due to time step restrictions caused by the stiffness of the equations which leads to prohibitive computational costs. In this work, we develop a new method for the full Euler system of gas dynamics based on partitioning the equations into a fast and a low scale. Such a method employs a time step which is independent of the speed of the pressure waves and works uniformly for all Mach numbers. Cell centered discretization on Cartesian meshes is proposed. Numerical results up to the three dimensional case show the accuracy, the robustness and the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

• The proposed method merges Monte Carlo approach and a finite volume method. • The statistical noise diminishes when the scaling parameter decreases. • The method is uniformly stable wrt the scaling parameter and the space mesh size. • No need of artificial transitions from the microscopic part to the macroscopic one. • Allows to run numerical tests on complex problems including full 3 dimensional ones. In this work, we develop a new class of numerical schemes for collisional kinetic equations in the diffusive regime. The first step consists in reformulating the problem by decomposing the solution in the time evolution of an equilibrium state plus a perturbation. Then, the scheme combines a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part, and is designed in such a way to be uniformly stable with respect to the diffusive scaling and to be consistent with the asymptotic diffusion equation. Moreover, since particles are only used to describe the perturbation part of the solution, the scheme becomes computationally less expensive – and is thus an asymptotically complexity diminishing scheme (ACDS) – as the solution approaches the equilibrium state due to the fact that the number of particles diminishes accordingly. This contrasts with standard methods for kinetic equations where the computational cost increases (or at least does not decrease) with the number of interactions. At the same time, the statistical error due to the Monte Carlo part of the solution decreases as the system approaches the equilibrium state: the method automatically degenerates to a solution of the macroscopic diffusion equation in the limit of infinite number of interactions. After a detailed description of the method, we perform several numerical tests and compare this new approach with classical numerical methods on various problems up to the full three dimensional case. [ABSTRACT FROM AUTHOR]