ON A DISCRETE FRAMEWORK OF HYPOCOERCIVITY FOR KINETIC EQUATIONS ALAIN BLAUSTEIN AND FRANCIS FILBET

2025-05-02 0 0 2.17MB 33 页 10玖币
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ON A DISCRETE FRAMEWORK OF HYPOCOERCIVITY FOR KINETIC
EQUATIONS
ALAIN BLAUSTEIN AND FRANCIS FILBET
Alain Blaustein
Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier
Toulouse, France
Francis Filbet
Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier
Toulouse, France
Abstract. We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equa-
tion written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally pre-
serves the stationary solution and the weighted L2relative entropy. Then, we adapt the arguments developed
in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of
the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with
respect to both the time variable and the scaling parameter at play.
Contents
1. Introduction 1
2. Hermite’s decomposition for the velocity variable 4
2.1. Main results 6
2.2. Preliminary results 8
2.3. Proof of Theorem 2.1 8
2.4. Proof of Theorem 2.2 12
3. Finite volume discretization for the space variable 15
3.1. Numerical scheme 15
3.2. Main results 17
3.3. Preliminary properties 18
3.4. Proof of Theorem 3.1 22
3.5. Proof of Theorem 3.2 24
4. Numerical simulations 26
4.1. Test 1 : centred Maxwellian 27
4.2. Test 2 : shifted Maxwellian 29
5. Conclusion and perspectives 29
Acknowledgement 32
References 32
1. Introduction
The Vlasov-Fokker-Planck equation is the kinetic description of the Brownian motion of a large system
of charged particles under the effect of an electric field. For example, in electrostatic plasma, where the
Coulomb force are taken into account, the time evolution of the electron distribution function fsolves the
2010 Mathematics Subject Classification. Primary: 82C40, Secondary: 65N08, 65N35 .
Key words and phrases. Hermite spectral method; Vlasov-Fokker-Planck; Hypocoercive estimates.
1
arXiv:2210.02107v1 [math.AP] 5 Oct 2022
Vlasov-Poisson-Fokker-Planck system, under the action of a self-consistent potential Φ:
∂f
∂t +v· ∇xf+qe
me
E· ∇vf=1
τe
divv(vf+T0∇vf),
−ε0∆Φ = qeZR3
fdv,
where ε0is the vacuum permittivity, qeand meare elementary charge and mass of the electrons, whereas
τeis the relaxation time due to the collisions of the particles with the surrounding bath.
Considering ε > 0 as the ratio between the mean free path of particles and the length scale of observation,
it allows to identify different regimes and the Vlasov equation may be written in a adimensional form
(1.1) ε∂f
∂t +v· ∇xf+E· ∇vf=ε
τ(ε)divv(vf+T0∇vf),
Our main purpose here is to build and analyse a numerical scheme able to capture two regimes of interest
for equation (1.1), in a linear framework: the long time behavior t→ ∞ and the diffusive regime ε→0. In
various situations, the scaling parameters at play may be non homogeneous across the system leading to
intricate situations, where both processes may coexist. Thus, we aim at designing a scheme robust enough
to capture simultaneously these different behaviors.
More precisely, we consider the one dimensional Vlasov-Fokker-Planck equation with periodic boundary
conditions in space, which reads
(1.2) ∂tf+1
ε(v ∂xf+E ∂vf) = 1
τ(ε)∂v(v f +T0∂vf),
with t≥0, position x∈Tand velocity v∈R, whereas the electric field derives from a potential Φ such
that E=−∂xΦ, with the following regularity assumption
(1.3) Φ ∈W2,∞(T).
We also define the density ρby integrating the distribution function in velocity,
(1.4) ρ(t, x) = ZR
f(t, x, v) dv.
It is worth to mention that there are already several works on preserving large-time behaviors of solutions
to the Fokker-Planck equation or related kinetic models. On the one hand, a fully discrete finite difference
scheme for the homogeneous Fokker-Planck equation has been proposed in the pioneering work of Chang
and Cooper [9]. This scheme preserves the stationary solution and the entropy decay of the numerical
solution. On the other hand, finite volume schemes preserving the exponential trend to equilibrium have
been studied for non-linear convection-diffusion equations (see for example [2,6,7,19]). More recently, in
[27], the authors investigate the question of describing correctly the equilibrium state of non-linear diffusion
and kinetic models for high order schemes. Let us also mention some works on boundary value problems
[14,8] where non-homogeneous Dirichlet boundary conditions are dealt with.
In the case of space non homogeneous kinetic equations, the convergence to equilibrium becomes tricky
because of the lack of coercivity since dissipation occurs only in the velocity variable whereas transport acts
in the space variable. Therefore, only few results are available and a better understanding of hypocoercive
structures at the discrete level is challenging. Let us mention a first rigorous work in this direction on the
Kolmogorov equation [28,17,18]. In [17], a time-splitting scheme is applied and it is shown that solutions
decay polynomially in time. In [28,18], a different approach has been used, based on the work of H´erau
[20] and Villani [31], for finite difference and a finite element schemes. Later, Dujardin, H´erau and Lafitte
[13] studied a finite difference scheme for the kinetic Fokker-Planck equation. Finally, in a more recent
work [5], the authors established a discrete hypocoercivity framework based on the continuous approach
provided in [12]. It is based on a modified discrete entropy, equivalent to a weighted L2norm involving
macroscopic quantities and the authors show quantitative estimates on the numerical solution for large
time and in the limit ε→0.
The present contribution can be considered as a continuation of this latter work in order to discretize
the kinetic Fokker-Planck equation with an applied force field. On the one hand, we consider the case
where the interactions associated to collisions and electrostatic effects have the same magnitude, that is,
τ(ε)∼ε, hence the limit t/ε →+∞corresponds to the long time behavior of equation (1.2). In this
2
regime, the distribution function frelaxes towards the stationary solution to the Vlasov-Fokker-Planck
equation ρ∞M, where the Maxwellian Mis given by
M(v) = 1
√2π T0
exp −|v|2
2T0,
whereas the density ρ∞is determined by
(1.5) ρ∞=c0exp −Φ
T0,
where the constant c0is fixed by the conservation of mass, that is,
ZT
ρ∞dx=ZZT×R
f0(x, v) dvdx .
Thus, we set f∞the stationary state of (1.2), defined as
f∞(x, v) = ρ∞(x)M(v)
and we expect that f→f∞as t/ε →+∞.
On the other hand, the diffusive regime corresponds to a frontier where collisions dominate but still
not enough to cancel completely the electrostatic effects. This situation occurs as ε→0 in the case
where τ(ε)∼τ0ε2, for some τ0>0. Due to collisions, the distribution of velocities also relaxes towards
a Maxwellian equilibrium. However, in this case, the spatial distribution converges to a time dependent
distribution ρwhose dynamics are driven by a drift-diffusion equation depending on the force field E.
Indeed, performing the change of variable x→x+τ0ε v in (1.2) and integrating with respect to v, we
deduce that the quantity
π(t, x) = ZR
f(t, x −τ0ε v, v) dv ,
solves the following equation
∂tπ+τ0∂xZR
E f (t, x −τ0ε v, v) dv−T0∂xπ= 0 .
According to its definition, πverifies: ρ∼πin the limit ε→0. Therefore, we may formally replace π
with ρand εwith 0 in the latter equation. This yields
f(t, x, v)−→
ε→0ρτ0(t, x)M(v),
where ρτ0solves
(1.6) ∂tρτ0+τ0∂x(E ρτ0−T0∂xρτ0) = 0 .
To be noted that this regime is an intermediate situation which contains more information than the long
time asymptotic since we have ρ→ρ∞by taking either t→+∞or τ0→+∞.
At the discrete level, Asymptotic-Preserving schemes have been developed to capture in a discrete setting
the diffusion limit, so that in the limit ε→0, the numerical discretization converges to the macroscopic
model (see for instance [23,26,22,25] on finite difference and finite volume schemes and [11,10] on particle
methods).
In the present article, our aim is to design a numerical scheme which is able to capture these two regimes
but also all the intermediate situations where ε2.τ(ε).ε. More precisely, we suppose that
(1.7) sup
ε>0
τ(ε)
ε≤τ0∈(0 ,+∞).
and distinguish two cases on τ(ε) :
(i) either the diffusive regime assumption
(1.8) τ(ε)
ε2−→
ε→0τ0<+∞,
where collisional effects strongly dominate;
3
(ii) or the intermediate regime assumption
(1.9) τ(ε)
ε2−→
ε→0+∞,
which may for instance correspond to τ(ε) = εβ, with 1 ≤β < 2. It describes all the intermediate
situations between long time and diffusive regime.
The starting point of our analysis is the following estimate, obtained multiplying equation (1.2) by
f / f∞, and balancing the transport term with the source term corresponding to the electric field thanks
to the weight f−1
∞
(1.10) 1
2
d
dtZTd×Rd|f−f∞|2f−1
∞dvdx+T0
τ(ε)ZT×R
∂vf
f∞
2
f∞dvdx= 0 .
This estimate is important since it yields a L2stability result on the solution to the Vlasov-Fokker-Planck
equation (1.2).
Our purpose is to design a numerical scheme for which such estimate occurs. To this aim, we split our
approach in two steps: we apply a spectral decomposition in velocity of fbased on Hermite decomposition
and we apply a structure preserving finite volume scheme for the space discretization. In the next section
(Section 2), we provide explicit convergence rates for the continuous model written in the Hermite basis (see
Theorems 2.1 and 2.2). This first step allows us to present the general strategy and to highlight the main
properties of the transport operator in order to design suitable numerical scheme. Therefore, in Section
3we adapt these latter results without any loss to the fully discrete setting using a structure preserving
finite volume scheme and an implicit Euler scheme for the time discretization (see Theorems 3.1 and 3.2).
The variety of situations that we aim to cover may lead to various and intricate behaviors. Therefore, we
successfully put great efforts into providing results which are uniform with respect to all parameters at
play: time t, scaling parameters (ε, τ0) and eventually the numerical discretization. The result is worth
the pain, since we propose in the Section 4various simulations, in which we are able to capture, at low
computational cost, a rich variety of situations.
2. Hermite’s decomposition for the velocity variable
The purpose of this section is to present a formulation of the Vlasov-Fokker-Planck equation (1.2) based
on Hermite polynomial and to provide quantitative results on fwhen ε→0 and t→+∞. These results are
identical to the ones obtained in the continuous case except that there are formulated on the corresponding
Hermite’s coefficients solution to a linear hyperbolic system. This formulation is well adapted to prepare
the fully discrete setting in Section 3.
We first use Hermite polynomials in the velocity variable and write the Vlasov-Fokker-Planck equation
(1.2) as an infinite hyperbolic system for the Hermite coefficients depending only on time and space. The
idea is to apply a Galerkin method only keeping a small finite set of orthogonal polynomials rather than
discretizing the distribution function in velocity [1,24]. The merit to use orthogonal basis like the so-called
scaled Hermite basis has been shown in [21,30,29] or more recently [16,4] for the Vlasov-Poisson system.
In this context the family of Hermite’s functions (Ψk)k∈Ndefined as
Ψk(v) = Hkv
√T0M(v),
constitutes an orthonormal system for the inverse Gaussian weight, that is,
ZR
Ψk(v) Ψl(v)M−1(v)dv=δk,l .
In the latter definition, (Hk)k∈Nstands for the family of Hermite polynomials defined recursively as follows
H−1= 0, H0= 1 and
ξ Hk(ξ) = √k Hk−1(ξ) + √k+ 1 Hk+1(ξ),∀k≥0.
Let us also point out that Hermite’s polynomials verify the following relation
H0
k(ξ) = √k Hk−1(ξ),∀k≥0.
4
Taking advantage of the latter relations, one can see why Hermite’s functions arise naturally when studying
the Vlasov-Poisson-Fokker-Planck model, especially in the diffusive regime, as they constitute an orthonor-
mal basis which diagonalizes the Fokker-Planck operator:
∂v[vΨk+T0∂vΨk] = −kΨk.
Therefore, we consider the decomposition of finto its components C= (Ck)k∈Nin the Hermite basis
(2.1) f(t, x, v) = X
k∈N
Ck(t, x) Ψk(v).
It’s worth to mention that we also may consider a truncated series neglecting high order coefficient in order
to construct a spectrally accurate approximation of fin the velocity variable.
As we have shown before, Hermite’s decomposition with respect to the velocity variable is a suitable
choice in our setting. When it comes to the space variable, we see from estimate (1.10) that the natural
functional framework here is the L2space with weight ρ−1
∞. Unfortunately, it is not very well adapted to the
space discretization since it may generate additional spurious terms difficult to control when dealing with
discrete integration by part. We bypass this difficulty by integrating the weight in the quantity of interest:
instead of working directly with f, we consider the quantity f / √ρ∞in order to get a well-balanced scheme
in the same spirit to what has been already done in [8,14] for well-balanced finite volume schemes. More
precisely, we set
Dk:= Ck
√ρ∞
in (2.1), and inject this ansatz in (1.2). Using that ρ∞E=T0∂xρ∞, we get that D= (Dk)k∈Nsatisfies
the following system
(2.2)
∂tDk+1
ε√kADk−1−√k+ 1 A?Dk+1=−k
τ(ε)Dk,
Dk(t= 0) = D0,ε
k,
where operators Aand A?are given by
Au= +pT0∂xu−E
2√T0
u ,
A?u=−pT0∂xu−E
2√T0
u .
In this framework, the equilibrium D∞to (2.2) is given by
(2.3) D∞,k =(√ρ∞,if k= 0 ,
0,else ,
and estimate (1.10) simply rewrites
(2.4) 1
2
d
dtkD(t)−D∞k2
L2+1
τ(ε)X
k∈N?
kkDk(t)k2
L2(T)= 0 ,
where k·kL2stands for the overall L2-norm with no weight
kDk2
L2=X
k∈NkDkk2
L2(T).
On top of that, the limit of the diffusive regime is given by Dτ0= (Dτ0,k)k∈Ndefined as follows
(2.5) Dτ0,k =
Dτ0,0,if k= 0 ,
0,else ,
where the first Hermite coefficient Dτ0,0solves the following drift-diffusion equation
(2.6) ∂tDτ0,0+τ0A?ADτ0,0= 0 ,
which is obtained substituting ρτ0with Dτ0,0√ρ∞in equation (1.6).
5
摘要：

ONADISCRETEFRAMEWORKOFHYPOCOERCIVITYFORKINETICEQUATIONSALAINBLAUSTEINANDFRANCISFILBETAlainBlausteinInstitutdeMathematiquesdeToulouse,UniversitePaulSabatierToulouse,FranceFrancisFilbetInstitutdeMathematiquesdeToulouse,UniversitePaulSabatierToulouse,FranceAbstract.Weproposeandstudyafullydiscrete n...

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