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In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov--Maxwell examples are presented to illustrate the good behavior of the approach. [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 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, 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]

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]

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]