Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series
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The Kruzhkov lectures 65
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Author information
Gregory A. Chechkin, Moscow Lomonosov State University, Russia.
E-mail: chechkin@mech.math.msu.su
Andrey Yu. Goritsky, Moscow Lomonosov State University, Russia.
E-mail: goritsky@mech.math.msu.su
Analytical and Numerical Aspects of Partial Differential Equations © de Gruyter 2009
Adaptive semi-Lagrangian schemes for Vlasov equations
Martin Campos Pinto
Abstract. This lecture presents a new class of adaptive semi-Lagrangian schemes – based on per-
forming a semi-Lagrangian method on adaptive interpolation grids – in the context of solving Vlasov
equations with underlying “smooth” flows, such as the one-dimensional Vlasov–Poisson system.
After recalling the main features of the semi-Lagrangian method and its error analysis in a uniform
setting, we describe two frameworks for implementing adaptive interpolations, namely multilevel
meshes and interpolatory wavelets. For both discretizations, we introduce a notion of good adap-
tivity to a given function and show that it is preserved by a low-cost prediction algorithm which
transports multilevel grids along any “smooth” flow. As a consequence, error estimates are estab-
lished for the resulting predict and readapt schemes under the essential assumption that the flow
underlying the transport equation, as well as its approximation, is a stable diffeomorphism. Some
complexity results are stated in addition, together with a conjecture of the convergence rate for the
overall adaptive scheme. As for the wavelet case, these results are new and also apply to high-order
interpolation.
Keywords. Fully adaptive scheme, semi-Lagrangian method, Vlasov equation, smooth characteristic
flows, adaptive mesh prediction, error estimates, interpolatory wavelets.
AMS classification. 65M12, 65M50, 82D10.
1 Introduction
In this lecture, we shall describe adaptive numerical methods for approximating Vlasov
equations, i.e., kinetic equations which model in statistical terms the nonlinear evolu-
tion of a collisionless plasma. In order to give the reader a specific example, we shall
somehow focus our presentation on the one-dimensional Vlasov–Poisson system
∂
t
f(t, x, v) + v · ∂
x
f(t, x, v) + E(t, x) · ∂
v
f(t, x, v) = 0, t > 0, x, v ∈ R,
∂
x
E(t, x) =
Z
R
f(t, x, v) dv − 1, f(0, ·, ·) = f
0
,
but we emphasize that our results actually apply to any nonlinear transport problem
associated with a smooth characteristic flow, “smooth” meaning here that the flow is a
Lipschitz diffeomorphism, see in particular Assumptions 3.2 and 3.3 below.
In order to save computational resources while approximating the complex and thin
structures that may appear in the solutions as time evolves, several adaptive schemes
have been proposed in the past few years, see, e.g., [3, 6, 7, 20], all based on the semi-
Lagrangian method originally introduced by Chio-Zong Cheng and Georg Knorr [9]
and later revisited by Eric Sonnendrücker, Jean Rodolphe Roche, Pierre Bertrand and
Alain Ghizzo [23]. A common feature of these new schemes lies in the multilevel,
70 Martin Campos Pinto
tree-structured discretization of the phase space, and a central issue appears to be the
prediction strategy for generating adaptive meshes that are optimal, i.e., that only retain
the “necessary” grid points.
Based on the regularity analysis of the numerical solution and how it gets trans-
ported by the numerical flow, it has recently been shown in [7] that a low-cost pre-
diction strategy could achieve an accurate evolution of the adaptive multilevel meshes
from one time step to the other, in the sense that the overall accuracy of the scheme was
monitored by a prescribed tolerance parameter ε representing the local interpolation er-
ror at each time step. In this lecture, we shall follow the same approach and propose
new algorithms for high-order wavelet-based schemes, together with error estimates.
The lecture is organized as follows. We shall start by presenting the Vlasov–Poisson
system in Section 2 and recall some important properties satisfied by its classical solu-
tions. The backward semi-Lagrangian method is then described in Section 3, together
with the main points of its error analysis, which yields a first convergence rate in the
case where the discretization is uniform. We detail next two distinct frameworks for
generating adaptive multilevel discretizations, namely adaptive meshes in Section 4
and interpolatory wavelets in Section 5. Finally, Section 6 is devoted to describing
algorithms that build multilevel grids which either are well-adapted to given functions
or remain well-adapted to transported functions. More precisely, for both discretiza-
tions we introduce a notion of ε-adaptivity to a given function and show that it is pre-
served by a low-cost prediction algorithm which transports multilevel grids along any
smooth flow. We end by proving new error estimates for the resulting semi-Lagrangian
schemes (with arbitrary interpolation order, under a central assumption on the approx-
imate flow) and state some partial complexity results.
In these notes, we shall often write A . B to express that A ≤ CB holds with a
constant C independent of the parameters involved in the inequality, A ∼ B meaning
that both A . B and B . A hold. As often as possible, we will nevertheless indicate
in the text the parameters on which these “invisible” constants may depend.
2 The Vlasov–Poisson system
The Vlasov system, as introduced by Anatoli
˘
ı Aleksandrovich Vlasov [24] in the late
1930s, describes in statistical terms the time evolution of rarefied plasmas, which are
gases of charged particles such as ions and electrons. In this section, we recall the form
of the system as well as its interpretation and describe some of its properties. Although
we shall mention the general equations in d = 3 physical dimensions, this lecture will
essentially focus on a simplified model in d = 1 physical dimension, leading to a two-
dimensional transport problem in the phase space. Most of our algorithms and proofs,
however, can be generalized to higher dimensions with no particular difficulty.
2.1 Description of the model
In the Vlasov model, the state of every species E of charged particles in the plasma is
represented at time t by a density function f
E
(t) defined in the phase space, i.e., the
Adaptive semi-Lagrangian schemes for Vlasov equations 71
subset of R
d
× R
d
which contains every possible position x and speed v. In particular,
the number of E-type particles that are located at time t in a physical domain ω
x
⊂ R
d
,
with speed v ∈ ω
v
⊂ R
d
, reads
Q
E
(t, ω) =
ZZ
ω
x
×ω
v
f
E
(t, x, v) dx dv. (2.1)
For the sake of simplicity, we shall assume that the effect of the magnetic field can be
neglected, which corresponds to an electrostatic approximation, and consider only two
species, namely:
•
positive (and heavy) ions, assumed to be uniformly distributed in space and time;
their density function will be denoted by f
p
(t, x, v) = f
p
(v), and we shall assume
that it is normalized, i.e.,
R
f
p
(v) dv = 1, and
•
much lighter electrons; their density function f is the main unknown of the model.
In dimension d = 1, the Vlasov–Poisson system reads as
∂
t
f(t, x, v) + v · ∂
x
f(t, x, v) + E(t, x) · ∂
v
f(t, x, v) = 0, (2.2)
∂
x
E(t, x) =
Z
R
f(t, x, v) dv − 1, (2.3)
where E represents the normalized, self-consistent electric field. In order to consider
the corresponding Cauchy problem, we supplement (2.2)–(2.3) with an initial condition
f(0, ·, ·) = f
0
(2.4)
assumed to be smooth (at least continuous) and compactly supported in R
2
.
2.2 Physical interpretation and characteristic flows
Since the end of the 18th century and the works of Charles-Augustin Coulomb, it is
known that a closed system of charged particles {q
i
∈ R, x
i
(t) ∈ R
3
}
i∈I
is subject to
binary interactions – the so-called Coulomb interactions – yielding an N-body prob-
lem. A less expensive model follows by considering the electromagnetic field which,
according to the theory that James Clerk Maxwell developed one century later, is cre-
ated by the particles and simultaneously influences their motion through the Lorentz
force. More precisely, if the associated charge and current densities are denoted by
ρ(t, x) :=
X
i∈I
q
i
δ
{x
i
(t)}
(x) and (t, x) :=
X
i∈I
v
i
(t)q
i
δ
{x
i
(t)}
(x),
respectively, where δ
x
i
(t)
stands for the Dirac mass located at x
i
(t) and v
i
(t) := x
0
i
(t) is
the particle’s speed, then the electromagnetic field (E, B)(t, x) created by the particles
72 Martin Campos Pinto
satisfies the following Maxwell system:
∇
x
· E = ρ/ε
0
, (2.5)
∇
x
× E = −∂
t
B, (2.6)
∇
x
· B = 0, (2.7)
∇
x
× B = µ
0
( + ε
0
∂
t
E). (2.8)
Here ∇
x
is the differential operator (∂
x
1
, ∂
x
2
, ∂
x
3
), × is the vector product in R
3
, and
ε
0
, µ
0
denote the permittivity and permeability constants. In turn, every particle is
subject to the Lorentz force
F
i
(t) = q
i
[E(x
i
(t)) + v
i
(t) × B(t, x
i
(t))]. (2.9)
According to Isaac Newton’s fundamental law of dynamics m
i
v
0
i
= F
i
, it follows that
every particle has a phase space trajectory that is a solution to the ordinary differential
equation
x
0
i
(t) = v
i
(t), v
0
i
(t) =
q
i
m
i
[E(x
i
(t)) + v
i
(t) × B(t, x
i
(t))]. (2.10)
The kinetic Vlasov model corresponds to a continuous limit of the above mean field
model when the particles are so many that each species can be represented by a smooth,
e.g., continuous, density function f
E
. In such a case, charge and current densities are
defined as
ρ(t, x) :=
X
E
q
E
Z
v∈R
3
f
E
(t, x, v)dv and (t, x) :=
X
E
q
E
Z
v∈R
3
vf
E
(t, x, v)dv,
and the electromagnetic field is again given by Maxwell’s system (2.5)–(2.8). As was
written above, we shall only consider two species E, namely positive ions and electrons.
The so-called Vlasov–Poisson system, which corresponds to an electrostatic approx-
imation, i.e., the effects of the magnetic field are neglected, can then be obtained by
writing the pointwise conservation of the electron density f along the characteristic
curves, which are defined as the solutions
t 7→ (X(t), V (t)) = (X(t; s, x, v), V (t; s, x, v)) (2.11)
to the ordinary differential system
∂
t
X(t) = V (t), ∂
t
V (t) = E(t, X(t)), (X, V )(s) = (x, v). (2.12)
Note that this is a natural extension of the trajectories (2.10) in the continuous limit. If
the electron density f satisfies
∂
t
f(t, X(t; 0, x, v), V (t; 0, x, v)) = 0 (2.13)
for any (x, v) in the phase space, one indeed finds
∂
t
f(t, x, v) + v · ∇
x
f(t, x, v) + E(t, x) · ∇
v
f(t, x, v) = 0, (2.14)
Adaptive semi-Lagrangian schemes for Vlasov equations 73
with an obvious definition for ∇
v
. Since E has then a vanishing curl, it derives from
an electric potential φ, i.e.,
E = −∇
x
φ, (2.15)
and equation (2.5) is equivalent to the Poisson equation
∆
x
φ = −
ρ
ε
0
, (2.16)
where ∆
x
≡ ∇
x
· ∇
x
denotes the Laplace operator. Equations (2.14)–(2.16) form the
Vlasov–Poisson system in three physical dimensions, from which the simplified model
(2.2)–(2.3) easily derives (with normalized constants) in one dimension.
Now, as long as E is bounded and continuously differentiable with respect to x, it
is known that the associated characteristic flow
F
t,s
: (x, v) 7→ (X, V )(t; s, x, v) (2.17)
is a measure preserving C
1
-diffeomorphism with inverse F
−1
t,s
= F
s,t
, see, e.g., [22].
Let us then notice that equation (2.13), which expresses the pointwise transport of f
along the characteristic curves, also yields a local transport property for the charge
density, in the sense that
ZZ
ω
f(0, x, v) dx dv =
ZZ
F
t,s
(ω)
f(t, x, v) dx dv for any ω ⊂ R
2d
, (2.18)
which is equivalent to saying that the flow is measure preserving. This property can be
seen as a consequence of the vanishing divergence of the field (x, v) 7→ (v, E(t, x)).
Indeed, it first allows to write (2.2) in a conservative form, i.e.,
∂
t
f(t, x, v) + ∇
x,v
· [(v, E(t, x))f (t, x, v)] = 0. (2.19)
Second, introducing the divergence-free field Φ(t, x, v) := (1, v, E(t, x)) defined on
D
τ,ω
= {(t, x, v) : t ∈ [0, τ ], F
−1
t,s
(x, v) ∈ ω} ⊂ [0, τ] × R
2d
, employing the Stokes
formula gives
ZZ
F
τ,s
(ω)
dx dv −
ZZ
ω
dx dv =
ZZ
∂D
τ,ω
Φ · n dσ =
ZZZ
D
τ,ω
∇
t,x,v
· Φ dt dx dv = 0,
where n denotes the outward unit vector normal to ∂D
τ,ω
. Here the first equality comes
from the fact that the boundary of D
τ,ω
is parallel to the field lines of Φ outside the
“faces” F
τ,s
(ω) and ω. As the latter equality precisely means that F
t,s
preserves the
Lebesgue measure, we see that its Jacobian determinant is equal to one. In particular,
property (2.18) readily follows from equation (2.13).
2.3 Existence of smooth solutions
According to the previous section, we shall say that (f, E) is a classical solution of the
Vlasov–Poisson system (2.2)–(2.4) if
74 Martin Campos Pinto
(i) f is continuous,
(ii) E is continuously differentiable,
(iii) the Vlasov equation (2.2) is satisfied in the sense of distributions.
Now, because the characteristic curves are defined on any point (x, v) of the phase
space, condition (iii) is equivalent to
(iii)’ f is constant along the characteristic curves defined by (2.12).
Remark 2.1. If f is only continuous, the derivatives appearing in (2.2) must be un-
derstood in the (weak) sense of distributions. Nevertheless, the solution is said to
be classical (or strong), because the characteristic curves are well-defined, and hence
equation (2.13) is satisfied in a classical sense.
One of the first results is due to Sergei Iordanskii (see [21]), who has proven in the
early 1960s global existence in time and uniqueness of classical solutions under certain
conditions. First, the initial datum f
0
must be continuous, and it must be integrable in
the following sense:
ρ
0
(x) =
Z
R
f
0
(x, v) dv − 1 < ∞ and
Z
R
v
2
θ(v) dv < ∞, (2.20)
where θ is a non-increasing function of |v| that dominates f
0
and f
p
(the density of the
positive ions). Second, the electric field must satisfy the limit condition
lim
x→−∞
E(t, x) = 0, t > 0, (2.21)
which is a reasonable condition since E is only defined up to a constant. Note that the
assumptions (2.20) are rather natural, since they allow to define the current and kinetic
energy densities,
(t, x) :=
Z
R
v[f(t, x, v) − f
p
(v)] dv and ε
k
(t, x) :=
Z
R
v
2
[f(t, x, v) + f
p
(v)] dv,
respectively, which are two fundamental physical quantities. Finally, Iordanskii shows
that f and E satisfy an additional equation, namely Ampère’s equation
∂
t
E(t, x) =
Z
R
v[f
p
(v) − f (t, x, v)] dv. (2.22)
In 1980, Jeffrey Cooper and Alexander Klimas [13] have extended these results to more
general limit conditions than (2.21), in particular they have addressed the periodic case
where x ∈ R/Z.
From now on, we shall only consider this case, moreover our initial data will always
be assumed to have a compact support in the phase space
Ω
∞
:= (R/Z) × R. (2.23)
Their result is the following.