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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]

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]

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]

This work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle-In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project. [ABSTRACT FROM AUTHOR]