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In this work, an exponential Discontinuous Galerkin (DG) method is proposed to solve numerically Vlasov type equations. The DG method is used for space discretization which is combined exponential Lawson Runge-Kutta method for time discretization to get high order accuracy in time and space. In addition to get high order accuracy in time, the use of Lawson methods enables to overcome the stringent condition on the time step induced by the linear part of the system. Moreover, it can be proved that a discrete Poisson equation is preserved. Numerical results on Vlasov-Poisson and Vlasov Maxwell equations are presented to illustrate the good behavior of the exponential DG method. • The main novelty of the paper is the development of high order exponential DG method for Vlasov type equations. • It involves the calculation of exponential of DG-matrices on this topic. • A discrete Poisson equation in DG frame work is given and satisfied. • These methods allow to derive high order accuracy in time, space and velocity. • These methods still ensure stability without the restrictive CFL type constraint coming from the linear part. [ABSTRACT FROM AUTHOR]

We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method, developed in Crouseilles et al. [Math. Models Methods Appl. Sci.23 (2017) 2031–2070] for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three dimensional Vlasov equation with an inhomogeneous, varying direction, strong magnetic field. Whenever the magnetic field has constant intensity, the oscillations generated by the stiff term are periodic. The homogenized model is then derived, and several state-of-the-art multiscale methods, in combination with the particle-in-cell discretization, are proposed for solving the Vlasov--Poisson equation. Their accuracy as much as their computational cost remain essentially independent of the strength of the magnetic field. The proposed schemes thus allow large computational steps, while the full gyro-motion can be restored by a linear interpolation in time. In the linear case, extensions are introduced for a general magnetic field (varying intensity and direction). Eventually, numerical experiments are exposed to illustrate the efficiency of the methods and some long-term simulations are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method solving the one space dimensional semiclassical kinetic graphene model introduced in [O. Morandi and F. Sch\"urrer, J. Phys. A, 44 (2012), pp. 265--301] involving fast oscillations in time, space, and momentum. This method can numerically capture the oscillatory space-time quantum solution pointwisely even without numerically resolving the frequency. We prove that the underlying micro-macro equations have smooth (up to a certain order of derivatives) solutions with respect to the frequency, and then we prove the uniform accuracy of the numerical discretization for a scalar model equation exhibiting the same oscillatory behavior. Numerical experiments verify the theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider uniformly accurate methods for solving the highly-oscillatory Vlasov equations with non-homogeneous magnetic field. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations by using a confinement property of the solution. A two-scale numerical scheme is proposed where the error estimates show that our scheme not only remains insensitive to the stiffness of the problem in terms of both accuracy and computational cost, but also preserves the confinement property at the discrete level. Our results are illustrated numerically on several examples. [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]

In this work, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov–Fokker–Planck system coupled with Poisson or Ampère equation. Splitting methods are used where the linear terms in space are treated by spectral or semi-Lagrangian methods and the nonlinear diffusion in velocity in the collision operator is treated using a stabilized Runge–Kutta–Chebyshev (RKC) integrator, a powerful alternative of implicit schemes. The new schemes are shown to exactly preserve mass and momentum. The conservation of total energy is obtained using a suitable approximation of the electric field. An H-theorem is proved in the semi-discrete case, while the entropy decay is illustrated numerically for the fully discretized problem. Numerical experiments that include investigation of Landau damping phenomenon and bump-on-tail instability are performed to illustrate the efficiency of the new schemes. • New stabilized Runge-Kutta methods for Vlasov-Fokker-Planck equation. • New second and fourth order discretizations of the Fokker-Planck operator. • Exact conservation of mass, momentum, and energy. • Proof of an H-theorem for the semi-discrete problem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider numerical methods for solving the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime. The model describes the behaviour of charged particles under a strong external magnetic field and the finite Larmor radius scaling. We discretise the equation under Particle-in-Cell method, where the characteristics equations are highly oscillatory system in the limit regime. We apply popular numerical integrators including splitting methods, multi-revolution composition methods, two-scale formulation method and limit solver to integrate the characteristics. We then highlight the strengths and drawbacks of each method. Finally, numerical experiments are presented, and comparisons on the accuracy, efficiency and long-time behaviour of the methods are made, pointing to the method with the best performance for this problem. [ABSTRACT FROM AUTHOR]

We introduce a new numerical strategy to solve a class of oscillatory transport partial differential equation (PDE) models which is able to capture accurately the solutions without numerically resolving the high frequency oscillations in both space and time. Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and other highly oscillatory waves. Our first main idea is to use the geometric optics ansatz, which builds the oscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman-Enskog expansion, for the new model. For a scalar model, we prove that so constructed models will have certain smoothness, and consequently, for a first-order approximation scheme we prove uniform error estimates independent of the (possibly small) wavelength. The method is extended to systems arising from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic phenomenon. Numerous numerical examples demonstrate that the method has the desired properties. [ABSTRACT FROM AUTHOR]

In this work, we focus on the numerical resolution of the four dimensional phase space Vlasov–Poisson system subject to a uniform strong external magnetic field. To do so, we consider a Particle-in-Cell based method, for which the characteristics are reformulated by means of the two-scale formalism, which is well-adapted to handle highly-oscillatory equations. Then, a numerical scheme is derived for the two-scale equations. The so-obtained scheme enjoys a uniform accuracy property, meaning that its accuracy does not depend on the small parameter. Several numerical results illustrate the capabilities of the method. [ABSTRACT FROM AUTHOR]

Ultrafast laser pulses interacting with plasmas can give rise to a rich spectrum of physical phenomena, which have been extensively studied both theoretically and experimentally. Less work has been devoted to the study of polarized plasmas, where the electron spin may play an important role. In this short review, we illustrate the use of phase-space methods to model and simulate spin-polarized plasmas. This approach is based on the Wigner representation of quantum mechanics, and its classical counterpart, the Vlasov equation, which are generalized to include the spin degrees of freedom. Our approach is illustrated through the study of the stimulated Raman scattering of a circularly polarized electromagnetic wave interacting with a dense electron plasma. [ABSTRACT FROM AUTHOR]

This work is devoted to the numerical simulation of a Vlasov--Poisson equation modeling charged particles in a beam submitted to a highly oscillatory external electric field. A numerical scheme is constructed for this model. This scheme is uniformly accurate with respect to the size of the fast time oscillations of the solution, which means that no time step refinement is required to simulate the problem. The scheme combines the forward semi-Lagrangian method with a class of uniformly accurate (UA) time integrators to solve the characteristics. These UA time integrators are derived by means of a two-scale formulation of the characteristics, with the introduction of an additional periodic variable. Numerical experiments are performed to show the efficiency of the proposed methods compared to conventional approaches. [ABSTRACT FROM AUTHOR]

In this work, which is the continuation of [SIAM J. Sci. Comput., 38 (2016), pp. A737-A764], we propose numerical schemes for linear kinetic equations which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake solving this stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context. [ABSTRACT FROM AUTHOR]

Various methods have been developed and tested over the years to solve the radiative transfer equation (RTE) with different results and trade-offs. Although the RTE is extensively used, the approximate diffusion equation is sometimes preferred, particularly in optically thick media, due to the lower computational requirements. Recently, multi-scale models, namely the domain decomposition methods, the micro–macro model and the hybrid transport–diffusion model, have been proposed as an alternative to the RTE. In domain decomposition methods, the domain is split into two subdomains, namely a mesoscopic subdomain where the RTE is solved and a macroscopic subdomain where the diffusion equation is solved. In the micro–macro and hybrid transport–diffusion models, the radiation intensity is decomposed into a macroscopic component and a mesoscopic one. In both cases, the aim is to reduce the computational requirements, while maintaining the accuracy, or to improve the accuracy for similar computational requirements. In this paper, these multi-scale methods are described, and the application of the micro–macro and hybrid transport–diffusion models to three-dimensional transient problems is reported. It is shown that when the diffusion approximation is accurate, but not over the entire domain, the multi-scale methods may improve the solution accuracy in comparison with the solution of the RTE. The order of accuracy of the numerical schemes and the radiative properties of the medium play a key role in the performance of the multi-scale methods. [ABSTRACT FROM AUTHOR]

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale ε . Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as ε → 0 , without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, it is known that such homogenization schemes are in general not accurate for both the highly oscillatory regime ε → 0 and the non-oscillatory regime ε ∼ 1 . In this paper, we introduce an Asymptotic Preserving method based on an exact micro–macro decomposition of the solution, which remains consistent for both regimes. [ABSTRACT FROM AUTHOR]

In this work, we propose some numerical schemes for linear kinetic equations in the anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation. Our aim is to develop numerical schemes for the original kinetic model which works for the different regimes, without being restricted by stability conditions of standard explicit time integrators. Starting from some numerical schemes for the diffusion asymptotics, their extension to the anomalous diffusion limit is then studied. In this case, it is crucial to capture the effect of the large velocities of the heavytailed equilibrium, so that some important transformations of the schemes derived for the diffusion asymptotics are needed. As a result, we obtain numerical schemes which enjoy the asymptotic preserving property in the anomalous diffusion limit; that is, they do not suffer from the restriction on the time step, and they degenerate towards the fractional diffusion limit when the mean free path goes to zero. We also numerically investigate the uniform accuracy and construct a class of numerical schemes satisfying this property. Finally, the efficiency of the different numerical schemes is shown through numerical experiments. [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 construct numerical schemes to solve kinetic equations with anomalous diffusion scaling. When the equilibrium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling should be made and the limit model is the so-called anomalous or fractional diffusion model. Our first scheme is based on a suitable micro–macro decomposition of the distribution function, whereas our second scheme relies on a Duhamel formulation of the kinetic equation. Both are Asymptotic Preserving (AP): they are consistent with the kinetic equation for all fixed value of the scaling parameter ε > 0 and degenerate into a consistent scheme solving the asymptotic model when ε tends to 0. The second scheme enjoys the stronger property of being uniformly accurate (UA) with respect to ε . The usual AP schemes known for the classical diffusion limit cannot be directly applied to the context of anomalous diffusion scaling, since they are not able to capture the important effects of large and small velocities. We present numerical tests to highlight the efficiency of our schemes. [ABSTRACT FROM AUTHOR]

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver. [ABSTRACT FROM AUTHOR]

• Introducing the Hamiltonian structure of the hybrid model to develop high-order splitting integrators. • Developing high order Lawson integrators combined with spectral and WENO methods in phase-space. • Adaptive time step strategy enables to reduce computational time while ensuring stability. • Comparison and validation on 1Dx-1Dv as well as in 1Dx-3Dv frameworks. In this work, we focus on the numerical approximation of a hybrid fluid-kinetic plasma model for electrons, in which energetic electrons are described by a Vlasov kinetic model whereas a fluid model is used for the cold population of electrons. First, we study the validity of this hybrid modeling in a two dimensional context (one dimension in space and one dimension in velocity) against the full (stiff) Vlasov kinetic model and second, a four dimensional configuration is considered (one dimension in space and three dimensions in velocity) following [1]. To do so, we consider two numerical Eulerian methods. The first one is based on the Hamiltonian structure of the hybrid system and the second approach, which is based on exponential integrators, enables to derive high order integrator and remove the CFL condition induced by the linear part. The efficiency of these methods, which are combined with an adaptive time stepping strategy, is discussed in the different configurations and in the linear and nonlinear regimes. [ABSTRACT FROM AUTHOR]

Abstract: This work is devoted to the numerical simulation of a Vlasov–Poisson model describing a charged particle beam under the action of a rapidly oscillating external field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a “two scale” reformulation of the initial equation, with the introduction of an additional periodic variable. [Copyright &y& Elsevier]

In this paper, we build a two-scale macro-micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circumscribed oscillations. The first of these functions has a shape which is close to the shape of the two-scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of two-scale asymptotic-preserving schemes. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional Euler equation with periodic boundary conditions. We construct time quasi-periodic solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction of propagation is orthogonal to this variable, and the support is concentrated on flat strips of the stationary state. The frequencies of the solution are given by the locally constant velocities associated with the stationary state. [ABSTRACT FROM AUTHOR]

Abstract: In this work, a new discretization scheme for the gyrokinetic quasi-neutrality equation is proposed. It is based on Isogeometric Analysis; the IGA which relies on NURBS functions, accommodates arbitrary coordinates and the use of complicated computation domains. Moreover, arbitrary high order degree of basis functions can be used and their regularity enables the use of a low number of elements. Here, this approach is successfully tested on elliptic problems like the quasi-neutrality equation arising in gyrokinetic models. In this last application, when polar coordinates are considered, a fast solver can be used and the non locality is dealt with a suitable decomposition which reduces the resolution of the gyrokinetic quasi-neutrality equation to a sequence of local 2D elliptic problems. [Copyright &y& Elsevier]

Abstract: Conservative methods for the numerical solution of the Vlasov equation are developed in the context of the one-dimensional splitting. In the case of constant advection, these methods and the traditional semi-Lagrangian ones are proven to be equivalent, but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity. In the non-constant advection case, they present an alternative to the traditional semi-Lagrangian schemes which can suffer from bad mass conservation, in this time splitting setting. [Copyright &y& Elsevier]

Abstract: A method for computing the numerical solution of Vlasov type equations on massively parallel computers is presented. In contrast with Particle In Cell methods which are known to be noisy, the method is based on a semi-Lagrangian algorithm that approaches the Vlasov equation on a grid of phase space. As this kind of method requires a huge computational effort, the simulations are carried out on parallel machines. To that purpose, we present a local cubic splines interpolation method based on a domain decomposition, e.g. devoted to a processor. Hermite boundary conditions between the domains, using ad hoc reconstruction of the derivatives, provide a good approximation of the global solution. The method is applied on various physical configurations which show the ability of the numerical scheme. [Copyright &y& Elsevier]

Abstract: Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov–BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests. [Copyright &y& Elsevier]

In this paper, we investigate the approximation of the solution to the Vlasov equation coupled with the Fokker–Planck–Landau collision operator using a phase space grid. On the one hand, the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed allowing to control spurious oscillations and preserving positivity and energy. On the other hand, the method preserves the main properties of the collision operators in order to reach the correct stationary state. Several numerical results are presented in one dimension in space and three dimensions in velocity. [Copyright &y& Elsevier]

Our purpose is to derive a model describing the evolution of particles at various scales following their kinetic energy. Fast particles will be described through a collisional kinetic equation of Boltzmann-BGK type. This equation will be coupled with a fluid model (Euler equations) that describes the evolution of slower particles. The main interest of this approach is to reduce the cost of numerical simulations. To cite this article: N. Crouseilles et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

• Stability analysis of exponential integrators and Lawson methods for hyperbolic problems • Constructive CFL condition • Validation on Vlasov-Poisson system • Validation on drift-kinetic system. The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g. , for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness. Our goal in this paper is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov–Poisson system and a drift-kinetic model of a ion temperature gradient instability. [ABSTRACT FROM AUTHOR]

In this work, a splitting strategy is introduced to approximate two-dimensional rotation motions. Unlike standard approaches based on directional splitting which usually lead to a wrong angular velocity and then to large error, the splitting studied here turns out to be exact in time. Combined with spectral methods, the so-obtained numerical method is able to capture the solution to the associated partial differential equation with a very high accuracy. A complete numerical analysis of this method is given in this work. Then, the method is used to design highly accurate time integrators for Vlasov type equations: the Vlasov--Maxwell and the Vlasov-HMF systems. Finally, several numerical illustrations and comparisons with methods from the literature are discussed. [ABSTRACT FROM AUTHOR]

In this paper, a bracket structure is proposed for the laser-plasma interaction model introduced in [19] , and it is proved by direct calculations that the bracket is Poisson which satisfies the Jacobi identity. Then splitting methods in time are proposed based on the Poisson structure. For the quasi-relativistic case, the Hamiltonian splitting leads to three subsystems which can be solved exactly. The conservative splitting is proposed for the fully relativistic case, and three one-dimensional conservative subsystems are obtained. Combined with the splittings in time, in phase space discretization we use the Fourier spectral and finite volume methods. It is proved that the discrete charge and discrete Poisson equation are conserved by our numerical schemes. Numerically, some numerical experiments are conducted to verify good conservations for the charge, energy and Poisson equation. [ABSTRACT FROM AUTHOR]