A numerical scheme for the quantum Fokker-Planck-Landau equation efficient in the fluid regime∗ Jingwei Hu†, Shi Jin† and Bokai Yan†
Abstract We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term QqF P L in order to remove the stiffness induced by the small Knudsen number. However, there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we propose to penalize QqF P L by the ‘classical’ linear Fokker-Planck operator. It is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. Numerical results for the Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both the fluid and kinetic regimes.
1
Introduction
The Fokker-Planck-Landau (FPL) equation is a kinetic model widely used in plasma physics. It describes the time evolution of charged particles in a plasma [18, 19]. When the quantum effects of particles are taken into account, for example, several bosons can occupy the same quantum state while only one fermion can occupy a particular quantum state, one has to use the following so-called quantum Fokker-Planck-Landau equation, ∂f 1 + v · ∇x f = QqF P L (f ), ∂t
x ∈ Ω ⊂ Rdx , v ∈ Rdv ,
(1.1)
where f (t, x, v) ≥ 0 is the phase space distribution function depending on time t, position x and particle velocity v. is the Knudsen number which measures the degree of rarefiedness of the particles. It is the ratio of the mean free path and the typical length scale. The quantum ∗ This
work was partially supported by NSF grant DMS-0608720 and NSF FRG grant DMS-0757285. SJ was also supported by a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the University of Wisconsin-Madison. † Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA (
[email protected],
[email protected],
[email protected]).
1
collision operator QqF P L is given by Z QqF P L (f )(v) = ∇v · A(v − v∗ ) [f∗ (1 ± θ0 f∗ )∇v f − f (1 ± θ0 f )∇v∗ f∗ ] dv∗
(1.2)
Rdv
with f = f (t, x, v) and f∗ = f (t, x, v∗ ). A(z) = Ψ(|z|)Π(z) is a dv × dv semi-positive definite matrix and Π(z) is the orthogonal projection onto the space orthogonal to z, Π(z) = I −
z⊗z , |z|2
I is the identity matrix.
(1.3)
For inverse-power law interactions, Ψ(|z|) = |z|γ+2 with −3 ≤ γ ≤ 1. The case γ = −3 refers to the Coulomb potential which is of primary importance in plasma applications. The parameter θ0 = ~dv , where ~ is the rescaled Planck constant. Here in (1.2) and the sequel, the upper sign will always correspond to the Bose gas (composed of bosons) while the lower sign to the Fermi 1 by the Pauli exclusion gas (composed of fermions). For the latter f must also satisfy f ≤ θ0 principle. Unlike the classical FPL equation, very few studies have been conducted on the quantum FPL equation. See [7] for a formal derivation from the quantum Boltzmann equation in the grazing collision limit and [20] for a spectral analysis of its linearization near the equilibrium. In the spatially homogeneous setting, the well-posedness and regularity of the solution were established in [1, 5] for Fermi-Dirac particles and the equilibrium states were rigorously determined in [2]. It is well-known that the equilibrium, in this context the quantum Maxwellian Mq (BoseEinstein or Fermi-Dirac distribution), is reached when the Knudsen number goes to zero. Then we could instead consider the limiting hydrodynamic equations satisfied by the moments of Mq . However, fluid equations are not adequate for many applications. Very often one has to deal with multiscale phenomena, where the Knudsen number varies between different regimes. Our goal in this paper is to design an efficient numerical scheme for the quantum FPL equation (1.1) that works uniformly for both kinetic and fluid regimes. The main difficulty arises when is very small: the right hand of equation (1.1) becomes stiff and due to the diffusive nature of the collision operator explicit schemes are subject to severe stability constraints (require time step ∆t = O(∆v 2 ), where ∆v is the mesh size in velocity domain). Implicit schemes don’t have such a restriction, but a fully nonlinear equation needs to be solved at each time step. Ideally, we would like an implicit scheme that allows larger time step and can be inverted easily. A class of asymptotic preserving schemes recently introduced in [9] has successfully resolved this issue for the classical Boltzmann equation. The basic idea is to penalize the collision term with a BGK operator: QcB = [QcB − λ (Mc − f )] + λ [Mc − f ], | {z } | {z } less stiff
(1.4)
stiff
where QcB is the classical Boltzmann collision operator. λ is some constant approximation of the spectrum of QcB . Mc is the classical Maxwellian given by Mc = ρ : density,
ρ (2πT )
dv 2
e−
u : macro-velocity,
2
(v−u)2 2T
,
T : temperature.
(1.5)
Now the first bracket in (1.4) is less stiff than the second one and can be treated explicitly. The second bracket will be discretized implicitly. Using the conservation properties of the BGK operator, this implicit term can actually be handled explicitly [6]. Later, a similar idea was applied to the quantum Boltzmann equation [8] and the classical FPL equation [14]. As the name implies, the quantum FPL equation (1.1) shares the features of both equations: (1) admits the quantum Maxwellian Mq as the equilibrium state like quantum Boltzmann; (2) becomes the classical FPL when θ0 → 0. Correspondingly, we are facing the numerical challenges from both sides: have to invert a nonlinear 2 by 2 system to define Mq [8]; the BGK operator is not suitable for penalization since QqF P L contains diffusion-like terms [14]. In this work, we propose to penalize (1.2) by a ‘classical’ Fokker-Planck (FP) operator:
with
QqF P L = [QqF P L − λPcF P (f, Mc )] + λPcF P (f, Mc )
(1.6)
f PcF P (f, Mc ) = ∇v · Mc ∇v . Mc
(1.7)
The idea is based on the observation that the classical Maxwellian (1.5), with the temperature replaced by the (quantum) internal energy, has the same first five moments as the quantum Maxwellian. In addition, the classical FP operator (1.7) has good conservation properties and is relatively easy to invert. Thus we arrive at a scheme uniformly stable in and avoid computing the complicated quantum Maxwellians. Furthermore, our numerical experiments show that the solution f will converge to Mq eventually within an error of O(∆t), no matter what the initial condition is. This guarantees the capturing of the fluid dynamic limit for small ∆t in the sense of the asymptotic-preserving schemes [12, 13]. To implement the above scheme, we need a fast and accurate solver for the quantum FPL operator QqF P L . The spectral method introduced in [23, 10] for the classical FPL operator can be easily extended to this case. Even though QqF P L is cubic, the fast algorithm still applies, allowing us to reduce the cost from O(n2 ) to O(n log n) (n = N dv , N is the number of discretized points in each direction of v). The details of the method are presented in the Appendix. The rest of the paper is organized as follows. In the next section, we briefly summarize the basic properties of the quantum FPL equation. In section 3, we describe the numerical schemes. Numerical examples are given in section 4 to illustrate the asymptotic property and the effectiveness of the scheme in both the kinetic and fluid regimes. Finally we make some concluding remarks in section 5.
2
The quantum FPL equation and its hydrodynamic limits
In this section we review some basic facts about the quantum FPL equation (1.1). They will be useful to design the numerical scheme. The weak form of the collision operator (1.2) reads: ZZ Z 1 T (∇v φ − ∇v∗ φ∗ ) A(v − v∗ )f∗ f (1 ± θ0 f∗ )(1 ± θ0 f ) QqF P L (f )φdv = − 2 d d d v v v R ×R R 1 ∇v f 1 ∇v∗ f∗ · − (2.1) dv∗ dv. 1 ± θ0 f f 1 ± θ0 f∗ f∗
3
Then
Z
Z
Z
QqF P L (f )dv = Rdv
QqF P L (f )vdv = Rdv
QqF P L (f )|v|2 dv = 0.
(2.2)
Rdv
Conservations of mass and momentum are straightforward. Conservation of energy follows from z T A(z) = (A(z)z)T = 0. f , one has If φ = ln 1 ± θ0 f Z QqF P L (f ) ln Rdv
T ∇v f 1 ∇v∗ f∗ 1 − A(v − v∗ ) 1 ± θ0 f f 1 ± θ0 f∗ f∗ Rdv ×Rdv ∇v f 1 ∇v∗ f∗ 1 − ·f∗ f (1 ± θ0 f∗ )(1 ± θ0 f ) dv∗ dv ≤ 0. 1 ± θ0 f f 1 ± θ0 f∗ f∗ (2.3)
1 f dv = − 1 ± θ0 f 2
ZZ
This is Boltzmann’s H-theorem. The last inequality comes from the semi-positivity of A(z). Moreover, Z f QqF P L (f ) ln dv = 0 ⇐⇒ QqF P L (f ) = 0 ⇐⇒ f = Mq , (2.4) 1 ± θ0 f Rdv where Mq is the quantum Maxwellian given by Mq =
1 1 . 2 θ0 z −1 e (v−u) 2T ∓1
(2.5)
µ
The macroscopic quantity z is called the fugacity: z = e T , µ is the chemical potential (see [11] for more details about the derivation of Mq ). (2.5) is the well-known Bose-Einstein (‘-’) and Fermi-Dirac (‘+’) distributions.
2.1
The hydrodynamic limit
Define the density ρ, macro-velocity u and specific internal energy e as Z Z Z 1 f |v − u|2 dv. ρ= f dv, ρ u = f vdv, ρe = 2 Rdv Rdv Rdv Then with f = Mq , the first dv + 2 moment equations of quantum Euler equations: ∂ρ + ∇x · (ρu) = 0, ∂t ∂(ρu) 2 + ∇x · ρu ⊗ u + ρeI = 0, ∂t dv ∂ 1 2 dv + 2 ρe + ρu + ∇x · ρe + ∂t 2 dv
(2.6)
(1.1) can be closed and yield the
(2.7) 1 2 ρu u = 0. 2
2 ρe holds for dv both classical and quantum gases. However, the intrinsic equipartition laws are quite different. dv For classical (monatomic) gas e = T , while for quantum gas ρ and e are connected with T 2 This form is exactly the same as the classical Euler equations. In fact pressure p =
4
and z (appeared in the definition of Mq (2.5)) by a nonlinear 2 × 2 system: dv (2πT ) 2 ρ = Q dv (z), 2 θ0
(2.8)
dv Q dv2+2 (z) T , e= 2 Q dv (z) 2
where Qν (z) denotes the Bose-Einstein function Gν (z) and Fermi-Dirac function Fν (z) respectively, Z ∞ 1 xν−1 Gν (z) = dx, 0 < z < 1, ν > 0; z = 1, ν > 1, (2.9) Γ(ν) 0 z −1 ex − 1 Z ∞ xν−1 1 dx, 0 < z < ∞, ν > 0, (2.10) Fν (z) = Γ(ν) 0 z −1 ex + 1 Z ∞ xν−1 e−x dx is the Gamma function. where Γ(ν) = 0
The physical range of interest for a Bose gas is 0 < z ≤ 1, with z = 1 corresponding to the degenerate case (the onset of Bose-Einstein condensation). z could be any positive real number for a Fermi gas and the degenerate case is reached when z 1. For small z, the integrand in (2.9) and (2.10) can be expanded in powers of z, Gν (z) = Fν (z) =
∞ X zn z2 z3 = z + + + ..., nν 2ν 3ν n=1 ∞ X
(−1)n+1
n=1
z2 z3 zn = z − ν + ν − .... ν n 2 3
(2.11) (2.12)
Thus both functions behave like z itself for z 1, and one recovers the classical limit. On the other hand, the first equation of (2.8) can be written as ρ
Q dv (z) = 2
where
(2πT )
dv 2
θ0 ,
(2.13)
ρ
dv is just the coefficient of the classical Maxwellian, and should be an O(1) quantity. (2πT ) 2 If θ0 → 0, then Q dv (z) → 0, which implies z → 0 by the monotonicity of function Qν . This is 2 consistent with the fact that one gets the classical FPL operator in (1.2) by letting θ0 → 0.
3
Numerical schemes
As discussed in the introduction, our goal is to design a numerical scheme for the quantum FPL equation (1.1) that works uniformly for different . Inspired by work [9], we seek a suitable operator P (f, Mq ) to penalize QqF P L . A first-order (in time) scheme should look like: λ f n+1 − f n 1 + v · ∇x f n = QqF P L (f n ) − λP f n , Mnq + P f n+1 , Mn+1 . q ∆t
(3.1)
To find f n+1 , we need to compute Mn+1 first. This can be accomplished by taking the moments q T 1 2 on both sides of (3.1), i.e. multiply by φ = 1, v, v and integrate with respect to v. If 2 5
P (f, Mq ) preserves mass, momentum and energy, one has Z U n+1 − U n + φ(v)v · ∇x f n dv = 0, (3.2) ∆t T 1 2 . Once we get U n+1 , Mn+1 is known, then we go back to (3.1) where U = ρ, ρu, ρe + ρu q 2 n+1 to compute f . From the above discussion, we naturally require the operator P (f, Mq ) to satisfy the following conditions: 1. It preserves mass, momentum and energy; 2. It is easy to invert; preferably linear in f ; 3. It contains diffusion-like terms to mimic the structure of QqF P L ; 4. It has a unique equilibrium solution: P (f, Mq ) = 0 ⇐⇒ f = Mq . Remark 3.1 The last condition is to insure the asymptotic property of the scheme. Simply speaking, if f n − Mnq = O(), then QqF P L (f n ) = O(), P f n , Mnq = O(). Thus (3.1) implies P f n+1 , Mn+1 = O(), so f n+1 − Mn+1 = O(). Later we will see this requirement is not q q necessary if we allow a weaker asymptotic condition. A similar problem has been thoroughly studied in [14] for the classical FPL equation and the FP operator (1.7) was suggested there as penalization. For the quantum Fokker-Planck-Landau equation, it is very natural to think of an operator P (f, Mq ) that depends on the quantum Maxwellian Mq and meets all of the above criteria. Even if one can find such an operator, there is still one difficulty left. In the classical case, if one knows U n+1 (conserved quantities), then Mn+1 is readily obtained, but things are not easy for Mn+1 . In fact, one has to invert the c q nonlinear system (2.8) to get z and T in order to define Mq [11, 8]. In practice this is very expensive (has to be done at every time step and every spatial point). Furthermore, setting a good initial guess for the iterative method is not trivial especially when z is close to the degenerate regime. Since PcF P works perfectly for the classical FPL equation and Mq is expensive to invert, motivated by our previous work on the quantum Boltzmann equation [8], we propose PcF P (1.7) as the penalization operator, where d2v dv 2 dv e− 4e (v−u) . (3.3) Mc = ρ 4πe The modification on Mc is just to replace the temperature T with internal energy e using relation dv e= T . The reason is because such defined Mc has the same first dv + 2 moments as Mq . 2 Therefore, our scheme for the quantum FPL equation (1.1) reads: f n+1 − f n 1 λ + v · ∇x f n = [QqF P L (f n ) − λPcF P (f n , Mnc )] + PcF P f n+1 , Mn+1 . c ∆t
(3.4)
The right hand side of (3.4) is still conservative, so one computes Mn+1 same as before. c It is important to notice that z and T are not present at all in this scheme. If they are desired variables for output, one only needs to convert between ρ, e and z, T through (2.8) at the final time, see [11]. 6
Remark 3.2 In general, one cannot expect f n+1 − Mn+1 = O() even if f n − Mnq = O() for q this new scheme. Since then QqF P L (f n ) = O(). The other terms on the numerator of the right hand side of (3.4) could be written as, roughly speaking, O (f n − Mnc ) + O f n+1 − Mn+1 , c n n n+1 n+1 n+1 n n+1 n which is equal to O f − Mq + O f − Mq + O Mq − Mq + O Mc − Mc . Assuming all the functions are smooth and ∆t > , the last two terms are of O(∆t), this implies O f n+1 − Mn+1 = O(∆t). q Remark 3.3 In [8] when the quantum BGK operator was penalized by the classical BGK operator, we were able to show n f − Mnq ≤ αn f 0 − M0q + O(∆t),
0 < α < 1.
(3.5)
Based on this simple analysis, we expect that f will converge to Mq in (3.4) eventually within an error of O(∆t) for any initial data. The numerical experiments in the next section actually confirm this property for our new scheme, but there is no theoretical justification yet.
3.1
The algorithm
We are ready to describe the detailed algorithm. n 1 at time tn , Given f n and U n = ρ, ρu, ρe + ρu2 2 Step 1 Approximate the transport term v · ∇x f n in (3.2) by a finite volume or finite difference method. Step 2 Compute U n+1 via (3.2) (using any quadrature rule for the integral). Step 3 Construct Mn+1 through (3.3). c Step 4 Evaluate QqF P L (f n ) by a fast spectral method presented in the Appendix. Step 5 Solve f n+1 by −1 λ∆t PcF P ·, Mcn+1 f n+1 = I − ∆t · f n − ∆tv · ∇x f n + (QqF P L (f n ) − λPcF P (f n , Mnc )) .
(3.6)
The discretization of PcF P and its inverse can be found in [14], where only a symmetric matrix needs to be inverted by the Conjugate-Gradient method. A remaining question is how to choose an appropriate λ to approximate the spectrum of QqF P L . The form (3.12) indicates a reasonable choice: λ should be the largest eigenvalue of matrix Z A(v − v∗ )f∗ (1 ± θ0 f∗ )dv∗ . (3.7) Rdv
Note that the integral is just the convolution of A and f (1 ± θ0 f ), which can be obtained via the Fourier transform.
7
3.2
Discussions on other possible penalty operators
An equivalent form of PcF P is written as v−u PcF P = ∇v · ∇v f + f . T
(3.8)
This reminds us of the quantum Fokker-Planck or Kaniadakis-Quarati equation [16, 17, 15], ∂f = ∇v · (∇v f + vf (1 ± θ0 f )) . ∂t
(3.9)
It was introduced as a simplified model of the full kinetic equation. The well-posedness and long-time asymptotics of the solution were investigated in [3, 4]. Recently it was also considered for modeling the Bose-Einstein condensate [25]. For the spatially inhomogeneous case, we define v−u PqF P = ∇v · ∇v f + f (1 ± θ0 f ) , (3.10) T analogous to its classical counterpart. Let’s check if this operator serves our purpose. It is not hard to see that PqF P satisfies conditions 3 and 4. However, it is not linear in f which needs more effort to invert than a linear one (although much better than QqF P L ). Most importantly, PqF P only preserves mass. Momentum and energy can be bounded [3], but are not strictly conserved. Another possibility is the following operator: P (f, Mq ) = ∇2v (f − Mq ).
(3.11)
This P clearly satisfies conditions 1-4, but it turns out to be a bad choice. We give a simple argument here. The quantum collision operator (1.2) can be rearranged as Z Z A(v − v∗ )∇v∗ f∗ dv∗ f (1 ± θ0 f ) . (3.12) ∇v · A(v − v∗ )f∗ (1 ± θ0 f∗ )dv∗ ∇v f − Rdv
Rdv
On the other hand, v−u Mq (1 ± θ0 Mq ) . ∇2v (f − Mq ) = ∇v · (∇v f − ∇v Mq ) = ∇v · ∇v f + T
(3.13)
Comparing (3.12) with (3.13) and (3.10), it is not surprising that the quantum FP operator (3.10) works better than the purely diffusive one (3.11). Similar scenario happens in the classical case, readers are referred to [14] for more explanations and numerical results regarding this issue.
4
Numerical examples
In all the simulations, the velocity space is assumed to be 2-D and the interactions between particles are the Coulomb interactions. A second order finite volume method with slope limiters [21] is applied to the transport part. Whenever fugacity z and temperature T are needed, we compute them as follows. System (2.8) (dv = 2) leads to Q21 (z) θ0 ρ = , (4.1) Q2 (z) 2π e 8
the left hand side is treated as one function of z, and inverted by the secant method. To evaluate the quantum function Qν (z), expansion (2.11) is used for the Bose-Einstein function. The FermiDirac function is computed by a direct numerical integration. The approach adopted is taken from [24] (Chapter 6.10).
4.1
Spatially homogeneous case
We first check the behavior of the solution in the spatially homogeneous case. Consider the nonequilibrium initial data ρ0 |v − u0 |2 |v + u0 |2 f0 (v) = exp − + exp − (4.2) 4πT0 2T0 2T0 with ρ0 = 1, T0 = 3/8 and u0 = (1, 1/2). Figure 1 shows the time evolution of f up to time t = 1.2 for a Bose gas. Here θ0 = 9 which corresponds to z = 0.6635 (the behavior of the quantum gas is significantly different from that of the classical gas). The Knudsen number = 1e − 4. The computational domain for v is taken as [−8, 8] × [−8, 8] with N = 64 in each direction. Under this condition, an explicit scheme would require ∆t = O(∆v 2 ) ≈ 1e − 6, while our scheme (3.4) gives fairly good results with a much coarser time step ∆t = 0.01. Furthermore, Figure 2 (left) clearly illustrates that the so obtained f is indeed driven to the quantum Maxwellian Mq rather than classical Maxwellian Mc . Figure 2 (right) is the evolution of the relative entropy H, Z 1 ± θ0 Mq f H= f ln . (4.3) 1 ± θ0 f Mq Rdv By the Boltzmann’s H-theorem, this quantity will decay to zero eventually.
4.2 4.2.1
Spatially inhomogeneous case Asymptotic property
We now numerically verify the asymptotic property of the scheme (3.4) for the spatially inhomogeneous equation. Consider the equilibrium initial data (I): f0 = Mq 0 , sin(2πx) + 2 cos(2πx) + 3 , u0 = (0, 0), T0 = , x ∈ [0, 1], (4.4) 3 4 and nonequilibrium initial data ρ0 |v − u0 |2 |v + u0 |2 (II): f0 = exp − + exp − , 4πT0 2T0 2T0 sin(2πx) + 2 cos(2πx) + 3 with ρ0 = , u0 = (0.2, 0), T0 = , x ∈ [0, 1]. (4.5) 3 4 We set θ0 = 4 to get a Bose gas that lies in the quantum regime (z : 0.24 ∼ 0.68). The Knudsen number = 1e − 4. v ∈ [−8, 8] × [−8, 8] with N = 64. Spatial size ∆x = 0.01. ∆t = 0.0013 by the CFL condition imposed on the transport part. The periodic boundary condition is used in the x-direction. In Figure 3, we report the time evolution of kf − Mq kL1 and kf − Mq kL2 for the equilibrium initial data (I) and nonequilibrium data (II) respectively. As one can see, the distances between f and Mq is about 1e − 3 ≈ O(∆t) as expected. For comparison, we also plot the norms of f − Mc , which are much bigger than those of f − Mq . with ρ0 =
9
t=0
t=0.1
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0 5
5 5
0 −5
5
0
0
0 −5
−5
−5
t=0.2
t=1.2
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0 5
5 5
0 −5
5
0
0
0 −5
−5
−5
Figure 1: The evolution of f for a Bose gas at times t = 0, 0.1, 0.2 and 1.2. θ0 = 9. = 1e − 4. ∆t = 0.01. v ∈ [−8, 8] × [−8, 8] on a 64 × 64 mesh. L1 norm
relative entropy H
1
1
10
10
||f−Mq||L1
0
0
||f−Mc||L1
10
10
−1
10
−1
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
10
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
−8
0
2
4
6
8
10 time
12
14
16
18
10
20
0
2
4
6
8
10 time
12
14
16
18
20
Figure 2: The time evolution of kf − Mq kL1 , kf − Mc kL1 (left) and the relative entropy H (right) for a Bose gas. θ0 = 9. = 1e − 4. ∆t = 0.01. v ∈ [−8, 8] × [−8, 8] with N = 64. 4.2.2
A shock tube problem
We next apply our scheme to the 1-D shock tube problem: 10
equilibrium initial data, Knudsen number=1e−4
nonequilibrium initial data, Knudsen number=1e−4
−1
−1
10
10
||f−Mc||L1
||f−Mc||L1 ||f−Mc||L2
||f−Mc||L2
−2
−2
10
10 ||f−Mq||L1
||f−Mq||L1
−3
−3
||f−Mq||L2
10
10
−4
−4
10
0
||f−Mq||L2
10 0.1
0.2
0.3
0.4
0.5 time
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 time
0.6
0.7
0.8
0.9
1
Figure 3: The distances between f and the Maxwellians for a Bose gas. Left: equilibrium initial data (I); right: nonequilibrium initial data (II). θ0 = 4. = 1e − 4. v ∈ [−8, 8] × [−8, 8] with N = 64. ∆x = 0.01. ∆t = 0.0013.
(ρl , ul , Tl ) = (1, 0, 1),
0 ≤ x ≤ 0.5,
0.5 < x ≤ 1.
(ρr , ur , Tr ) = (0.125, 0, 0.25),
(4.6)
We adjust θ0 (= 4) to get a Fermi gas in the quantum regime (zl = 0.8901, zr = 0.3748). The Knudsen number = 1e − 4. v ∈ [−8, 8] × [−8, 8] with N = 64. ∆x = 0.01. ∆t = 0.0013 as before. In this hydrodynamic regime, the solution of the kinetic equation should be close to that of the fluid dynamic equations. So we compare the results of scheme (3.4) with the kinetic scheme (KFVS scheme in [11]) for the quantum Euler equations (2.7). Figure 4 shows the macroscopic quantities ρ, u, z and T at time t = 0.2. 4.2.3
Mixing regime
So far the Knudsen number has been fixed in the numerical simulation, and we’ve seen that our scheme works well in the fluid regime. In the real situation the Knudsen number usually varies between different regimes. The next example is devoted to this kind of problem. Assume is space-dependent: = 0 + 0.05(tanh(5 − 10x) + tanh(5 + 10x)),
0 = 1e − 3,
x ∈ [0, 1],
(4.7)
as shown in Figure 5. We consider a Fermi gas with θ0 = 2.25 (z : 0.16 ∼ 0.76) starting from the equilibrium initial data (4.4). Let v ∈ [−8, 8] × [−8, 8] with N = 64. ∆x = 0.01. ∆t = 0.0013 by the CFL condition, independent of . Periodic boundary condition is applied to the x-direction. The macroscopic quantities ρ, u and T at time t = 0.2 are plotted in Figure 6. The reference solution is computed by an explicit second-order Runge-Kutta method with a much smaller time step ∆t = 6.7771e − 06. All the results agree pretty well which demonstrates that the scheme is also reliable for problems with mixing scales. 11
u1
ρ
1.1
1 1 0.9 0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4 0.3
0.4
0.2
0.3
0.1 0.2 0 0.1
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
z
0.5 x
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
T
1
1.2 1.1
0.9
1 0.8 0.9 0.7
0.8
0.6
0.7 0.6
0.5
0.5 0.4 0.4 0.3
0.2
0.3
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
0.2
1
0
0.1
0.2
0.3
0.4
0.5 x
Figure 4: Fermi gas in the fluid regime = 1e − 4. θ0 = 4. Density ρ, velocity u1 , fugacity z and temperature T at t = 0.2. v ∈ [−8, 8] × [−8, 8] with N = 64, ∆x = 0.01, ∆t = 0.0013. ◦ : scheme (3.4) for quantum FPL equation; solid line: KFVS scheme for quantum Euler equations. ε 0.1
ε≈0.1
0.08
0.06
0.04
0.02
ε≈0.001 0 0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
Figure 5: Space-dependent .
5
Conclusion
An efficient numerical scheme was constructed for the quantum Fokker-Planck-Landau equation (1.1) that works uniformly for different Knudsen numbers. The main idea is to penalize the 12
ρ
0.9
u1
0.4
0.85
T
1.1 1.05
0.3
1
0.8
0.2 0.95
0.75
0.1
0.9
0.7
0
0.85
0.65 0.8
−0.1 0.6
0.75
−0.2
0.55
0.7
−0.3
0.5 0.45
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
−0.4
0.65
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
Figure 6: Fermi gas in the mixing regime. θ0 = 2.25. Density ρ, velocity u1 and temperature T at t = 0.2. v ∈ [−8, 8] × [−8, 8] with N = 64. ∆x = 0.01. ◦: scheme (3.4) with ∆t = 0.0013; solid line: explicit second-order Runge-Kutta scheme with ∆t = 6.7771e − 06. quantum collision operator QqF P L (1.2) by the ‘classical’ Fokker-Planck operator PcF P (1.7), with the temperature T replaced by internal energy e in the classical Maxwellian Mc . The implicit PcF P term can be easily inverted while QqF P L is much more complex to invert. A remarkable feature of the scheme is that the numerical solution f is still driven to the quantum Maxwellian rather than Mc under the fluid regime. We also developed a fast spectral method for the collision operator QqF P L following its classical counterpart.
Appendix A
A fast spectral method for the quantum FPL collision operator
In this Appendix, we outline the fast spectral method for evaluating the quantum FPL collision operator (1.2). It is an extension of the spectral method in [23, 10] for the classical FPL operator. In order to develop a spectral approximation, we restrict the function f on the cube DT = [−T, T ]dv , and extend it periodically to the whole domain. T is chosen such that T = 2R, where B(0, R) is the compact support of f [22]. We can then write the operator (1.2) as [23], Z QqF P L (f )(v) = ∇v ·
A(g) [h(v + g)∇v f (v) − h(v)∇g f (v + g)] dg,
(A.1)
B(0,2R)
gg T where A(g) = |g|γ+2 I − 2 , h(v) = f (v)(1 ± θ0 f (v)). |g| Now f and h are approximated by truncated Fourier series, N 2
f (v) ≈
−1 X
π fˆk ei T k·v ,
fˆk =
k=− N 2
13
1 (2T )dv
Z DT
π
f (v)e−i T k·v dv;
(A.2)
N 2
−1 X
h(v) ≈
ˆ k ei Tπ k·v , h
ˆk = h
k=− N 2
Z
1 (2T )dv
π
h(v)e−i T k·v dv.
(A.3)
DT
ˆ qF P L , Plugging them into (A.1), one can get the k-th mode of Q
2
ˆ qF P L,k = − π Q T2
N 2
−1 X
"Z
π ˆm |g|γ+2 [l(l + m) − (l · µ)((l + m) · µ)]ei T m·g dg fˆl h
B(0,2R)
l,m=− N 2 l+m=k
#
Z
γ+2
−
|g|
π m·g iT
[m(l + m) − (m · µ)((l + m) · µ)]e
ˆl , dg fˆm h
B(0,2R)
(A.4) where µ =
g . Following [23], we define |g| Z π F (m) = |g|γ+2 ei T m·g dg, B(0,2R) Z π Ipq (m) = |g|γ gp gq ei T m·g dg,
(A.5) p, q = 1, . . . dv .
(A.6)
B(0,2R)
Then (A.4) can be recast as 2
ˆ qF P L,k = − π Q T2
N 2
−1 h X
ˆ m + mlF (m)fˆl h ˆ m − m2 F (m)fˆm h ˆ l − mlF (m)fˆm h ˆl l2 F (m)fˆl h
l,m=− N 2 l+m=k
−
dv X
ˆm − lp lq Ipq (m)fˆl h
p,q=1
+
dv X
dv X
ˆm lp mq Ipq (m)fˆl h
p,q=1
ˆl + mp lq Ipq (m)fˆm h
p,q=1
dv X
# ˆl . mp mq Ipq (m)fˆm h
(A.7)
p,q=1
Every term in (A.7) is a convolution, which can be computed by the Fast Fourier Transform in O(n log(n)) operations, where n = N dv is the total points in the velocity space. As for F (m) and Ipq (m), we precalculated them according to the formulas given in [23]. In γ+4 needs to be multiplied to their results since we are dealing with the 2-D case, a factor (T /π) the arbitrary domain DT .
References [1] V. Bagland. Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Proc. R. Soc. Edinburgh, Ser. A, 134:415–447, 2004. [2] V. Bagland and M. Lemou. Equilibrium states for the Landau-Fermi-Dirac equation. Banach Center Publ., 66:29–37, 2004. [3] J. A. Carrillo, P. Lauren¸cot, and J. Rosado. Fermi-Dirac-Fokker-Planck equation: well posedness and long-time asymptotics. J. Differ. Equ, 247:2209–2234, 2009. 14
[4] J. A. Carrillo, J. Rosado, and F. Salvarani. 1D nonlinear Fokker-Planck equations for fermions and bosons. Appl. Math. Lett., 21:148–154, 2008. [5] Y. Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinet. Relat. Models, 3:645–667, 2010. [6] F. Coron and B. Perthame. Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal., 28:26–42, 1991. [7] P. Danielewicz. Nonrelativistic and relativistic Landau/Fokker-Planck equation for arbitrary statistics. Phys. A, 100:167–182, 1980. [8] F. Filbet, J. W. Hu, and S. Jin. A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime. preprint. [9] F. Filbet and S. Jin. A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys., 229:7625–7648, 2010. [10] F. Filbet and L. Pareschi. A numerical method for the accurate solution of the FokkerPlanck-Landau equation in the nonhomogeneous case. J. Comput. Phys., 179:1–26, 2002. [11] J. W. Hu and S. Jin. On kinetic flux vector splitting schemes for quantum Euler equations. Kinet. Relat. Models, 4:517–530, 2011. [12] S. Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput., 21:441–454, 1999. [13] S. Jin. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review (Lecture Notes for Summer School on “Methods and Models of Kinetic Theory”, ˜ di Parma, to appear. Porto Ercole, June 2010). Rivista di Matematica della UniversitA [14] S. Jin and B. Yan. A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation. J. Comput. Phys., to appear. [15] G. Kaniadakis. Generalized Boltzmann equation describing the dynamics of bosons and fermions. Phys. Lett. A, 203:229–234, 1995. [16] G. Kaniadakis and P. Quarati. Kinetic equation for classical particles obeying an exclusion principle. Phys. Rev. E, 48:4263–4270, 1993. [17] G. Kaniadakis and P. Quarati. Classical model of bosons and fermions. Phys. Rev. E, 49:5103–5110, 1994. [18] L. D. Landau. Die kinetische Gleichung f¨ ur den Fall Coulombscher Wechselwirkung. Phys. Z. Sowjetunion, 10:154–164, 1936. [19] L. D. Landau. The kinetic equation in the case of Coulomb interaction. Zh. Eksper. i Teoret. Fiz., 7:203–209, 1937. [20] M. Lemou. Linearized quantum and relativistic Fokker-Planck-Landau equations. Math. Meth. Appl. Sci., 23:1093–1119, 2000.
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[21] R. J. LeVeque. Numerical Methods for Conservation Laws. Birkh¨auser Verlag, Basel, second edition, 1992. [22] L. Pareschi and G. Russo. Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal., 37:1217–1245, 2000. [23] L. Pareschi, G. Russo, and G. Toscani. Fast spectral methods for the Fokker-Planck-Landau collision operator. J. Comput. Phys., 165:216–236, 2000. [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, third edition, 2007. [25] G. Toscani. Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles. preprint.
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