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2. THE VLASOV-POISSON SYSTEM
21
Whence get X(t; s, x, v) = x cos(t−s)+v sin(t−s) and V (t; s, x, v) = −x sin(t−
s) + v cos(t − s) form which we compute the solution
f (x, v, t) = f0 (x cos t − v sin t, x sin t + v cos t).
2. The Vlasov-Poisson system
2.1. The equations. The Poisson equation is obtained from the Maxwell equations, when the electric and magnetic fields are not, or only very little, time dependent.
We then get the stationary Maxwell equations
(20)
(21)
(22)
(23)
∇×
∇×
∇·
∇·
B
E
E
B
=
=
=
=
J,
0,
ρ,
0.
In this case the electric and magnetic fields are decoupled, and in many cases, because
B itself is small, or because its contribution in the Lorentz force v × B is small, we
shall only need the electric field, which is given by equations (21) and (22). Equation
(21) implies that E derives from a scalar potential E = −∇φ, and then (22) implies the
Poisson equation
−∆φ = ρ,
along with adequate boundary conditions.
We consider the dimensionless Vlasov-Poisson equation for one species with a neutralizing background
∂f
+ v · ∇x f − E · ∇v f = 0,
(24)
∂t
(25)
−∆φ = 1 − ρ,
with
E = −∇φ,
Z
ρ(x, t) =
f (x, v, t) dv.
The domain on which the system is posed is considered periodic in x and the whole
space R3 in velocity.
We first notice that the Vlasov equation (24) can also be written in conservative
form
∂f
(26)
+ ∇x,v · (Ff ) = 0,
∂t
with F = (v, −E)T such that ∇x,v · F = 0.
2.2. Conservation properties. The Vlasov-Poisson system has a number of conservation properties that need special attention when developing numerical methods. In
principle it is beneficial to retain the exact invariants in numerical methods and when
it is not possible to keep them all as is the case here, they can be use to monitor the
validity of the simulation by checking that they are approximately conserved with good
accuracy.
Proposition 5. The Vlasov-Poisson system verifies the following conservation properties:
• Maximum principle
(27)
0 ≤ f (x, v, t) ≤ max(f0 (x, v)).
(x,v)