Motivated by the difficulty arising in the numerical simulation of the movement of charged particles in presence of a large external magnetic field, which adds an additional time scale and thus imposes to use a much smaller time step, we perform in this paper a homogenization of the Vlasov equation and of the Vlasov-Poisson system which yield approximate equations describing the mean behavior of the particles. The convergence proof is based on the two scale convergence tools introduced by N'Guetseng and Allaire. We also consider the case where, in addition to the magnetic field, a large external electric field orthogonal to the magnetic field and of the same magnitude is applied.
In many kind of devices involving charged particles, like electron guns, diodes or tokamaks, a large external magnetic field needs to be applied in order to keep the particles on the desired tracks. In Particle-In-Cell (PIC) simulations of such devices, this large external magnetic field obviously needs to be taken into account when pushing the particles. However, due to the magnitude of the concerned field this often adds a new time scale to the simulation and thus a stringent restriction on the time step. In order to get rid of this additional time scale, we would like to find approximate equations, where only the gross behavior implied by the external field would be retained and which could be used in a numerical simulation.
The trajectory of a particle in a constant magnetic field B is a helicoid along the magnetic field lines with a radius proportional to the inverse of the magnitude of B. Hence, when this field becomes very large the particle gets trapped along the magnetic field lines. However due to the fast oscillations around the apparent trajectory, its apparent velocity is smaller than the actual one. This result has been known for some time as the “guiding center” approximation, and the link between the real and the apparent velocity is well known in terms of B. We refer to Lee [13] and Dubin et al [5] for a complete physical viewpoint review on this subject. In the case of a cloud of particles, the movement of which is described by the Vlasov-Poisson equations, the situation is less clear as in the one particle case because of the non linearity of the problem. In particular, the question of knowing if the mutual influence of the particles can be expressed in terms of their apparent motion or if the oscillation generates additional effects is important.
In this paper, we deduce the “guiding center” approximation in the framework of partial differential equations via an homogenization process on the linear Vlasov equation. Then, we show that in a cloud of particles the mutual influence of the particles can be expressed in term of their apparent motion without any additionnal terms. This is provided applying an homogenization process on the Vlasov-Poisson system, similar to the one used for the linear Vlasov equation and using the regularity of the charge density.
Hence, we apply first a homogenization process to the linear Vlasov equation with a strong and constant external magnetic field. In other words for a constant vector M ∈ I S 2 , we consider the following equation:
(1.1)
In this equation f ε ≡ f ε (t, x, v) with t ∈ [0, T ), for any T ∈ IR + , x ∈ IR 3
x and v ∈ IR 3 v . For convenience, we introduce the notations Ω = IR 3
x × IR 3 v , O = [0, T ) × IR 3
x and Q = [0, T ) × Ω. The initial data satisfies
The electric field E ε ≡ E ε (t, x) and the magnetic field B ε ≡ B ε (t, x) are defined on O, both bounded in L ∞ (0, T ; L 2 (IR 3
x )) and satisfy
and
for any T ∈ IR + . With those conditions, for every ε there exists a unique solution of the equation (1.1)
We characterize here the equation satisfied by the limit f (in some weak topologies) of (f ε ) ε .
The first main result of the paper is the following:
Moreover, denoting for any vector v, v = (v • M)M, f is the unique solution of:
where u(v, τ ) is the rotation of angle τ around M applied to v (see (2.7) for more details).
The deduction of this Theorem uses the notion of two scale convergence introduced by N’Guetseng [15] and Allaire [2]. This convergence result is the following:
Theorem 1.2 (N’Guetseng [15] and Allaire [2]) If a sequence f ε is bounded in L ∞ (0, T ; L 2 (Ω)), then for every period θ, there exists a θ-periodic profile
) such that for all ψ θ (t, τ, x, v) regular, with compact support with respect to (t, x, v) and θperiodic with respect to τ we have, up to a subsequence,
Above, L ∞ θ (IR τ ) stands for the space of functions being L ∞ (IR) and being θ-periodic and ψ ε θ ≡ ψ θ (t, t ε , x, v). The profile F θ is called the θ-periodic two scale limit of f ε and the link between F θ and the weak- * limit f is given by
(1.7)
This result has been used with success in the context of homogenization of transport equation with periodically oscillating coefficients by E [6] and Alexandre and Hamdache [1] and in the context of kinetic equations with strong and periodically oscillating coefficients in Frénod [7] and Frénod and Hamdache [8].
Here, since the strong magnetic field induces periodic oscillations in f ε , the two scale limit describes well its asymptotic behavior. Therefore, it is the right tool to tackle homogenization of equation (1.1), and Theorem 1.1 is a consequence of the following result concerning the two scale limit of f ε .
where u(v, τ ) is the rotation of angle τ around M applied to v (see (2.7) for more details).
Remark 1.1 This last Theorem can be generalised to the case of a non uniform strong magnetic field, see Remark 2.2.
The method we develop enables also to deduce the
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