Asymptotic-Preserving Exponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy

Transcription

1 J Sci Comput () 6: 74 DOI.7/s Asymptotic-Preserving Eponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy Jingwei Hu Qin Li Lorenzo Pareschi Received: 3 October 3 / Revised: 3 May 4 / Accepted: 8 May 4 / Published online: 6 May 4 Springer Science+Business Media New York 4 Abstract In this paper we develop high order asymptotic preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi (J Comput Phys 9:4 4, 4) where asymptotic preserving eponential Runge Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian steady states characterizing the quantum kinetic behavior. We show that the proposed schemes achieve high-order accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen number ranging from kinetic regime to fluid regime. Computational results are presented for both Bose gas and Fermi gas. Keywords Quantum Boltzmann equation Asymptotic preserving methods Eponential Runge Kutta schemes Mathematics Subject Classification 6L4 6L6 3Q 8C This work was partially supported by RNMS-7444 (KI-Net) and by PRIN 9 project Advanced numerical methods for kinetic equations and balance laws with source terms. J. Hu Institute for Computational Engineering and Sciences (ICES) and Bureau of Economic Geology (BEG), The University of Teas at Austin, East 4th St, Stop C, Austin, TX 787, USA hu@ices.uteas.edu Q. Li (B) Department of Computing + Mathematical Sciences (CMS), The Annenberg Center, California Institute of Technology, E. California Blvd., Pasadena, CA 9, USA qinli@caltech.edu L. Pareschi Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 3, 44 Ferrara, Italy lorenzo.pareschi@unife.it 3

2 6 J Sci Comput () 6: 74 Introduction The quantum Boltzmann equation (QBE), also known as the Nordheim-Uehling-Uhlenbeck equation, describes the non-equilibrium dynamics of a dilute quantum gas consisting of elementary particles of bosons or fermions [3]. With the quantum information embedded in the collision term, this equation gives better description to quantum particles, and it recovers the standard Boltzmann equation for the classical particles in the zero limit of the Planck constant, and thus covers a wider range of particle behavior. The QBE and its variants have many applications in science and engineering, including the kinetic description of Bose Einstein condensate [3,33], and the modeling of electron interactions in semiconductor devices [8,4]. In this paper we design a class of high order numerical methods for quantum Boltzmann equation, that is accurate and efficient in both kinetic and hydrodynamic regimes for all Planck constants. In kinetic theory, the time discretization represents a computational challenge in the construction of numerical methods, especially in stiff regimes, when the collisional scale becomes dominant over the transport of particles, and the fluid-dynamic limit is achieved. To resolve the collision term, the time step is severely controlled by the Knudsen number for numerical stability if eplicit schemes are used. On the other hand, though the implicit schemes allows larger time steps for stability, they are impractical due to the nonlinear and nonlocal collision operator. Many techniques have been developed to address such issues in recent years, and we specifically mention the micro-macro decomposition [], the BGK penalization method [9], and the eponentialrunge Kutta methods [6,]. The feature shared among these techniques is that the schemes are unconditionally stable, capturing the asymptotic limits automatically with time step relaed from the constraints of the Knudsen number, and therefore, numerically less complicated than other possible approaches, for eample, the domain decomposition strategies and hybrid methods at different levels [4,,7,34]. For a nice survey on asymptoticpreserving (AP) scheme for various kinds of systems see, for instance, the review paper by Jin [7]. In the case of Boltzmann-type kinetic equations we refer to a recent review by Pareschi and Russo [9]. In this work we etend the asymptotic preserving eponential Runge Kutta method developedin [6,] to the quantum Boltzmann equation. The etension to the multi-species Boltzmann equation was done in []. We refer the reader to [3] for an introduction to time integration eponential techniques. The discussion on AP property and stability can be found in [6,8]. New difficulties in the quantum case would be: The steady states are not classical Mawellian (Gaussian distribution) and to obtain the local equilibrium the quantum Mawellian (Bose Einstein or Fermi Dirac distribution), a nonlinear system needs to be inverted; The methods developed need to be uniformly high order and efficient for all Planck constants, and capture the classical limit. An asymptotic-preserving method for the quantum Boltzmann equation has been proposed in [8], where a first-order IMEX scheme combined with the standard BGK penalization idea was used. In particular, the classical Mawellian was suggested in [8] as an alternative to the complicated quantum Mawellian for penalty. This replacement saves fairly amount of computational cost, but the price to pay is the loss of the strong AP property (namely, in the fluid-limit the distribution function should converge in one time step to its physical equilibrium state). Moreover, as the scheme is of IMEX type, it is hard to etend the method to very high order [7]. In comparison, our new schemes possess the strong AP property and, in 3

3 J Sci Comput () 6: 74 7 principle, could achieve arbitrarily high order. As we shall see, the presence of non classical steady states has a profound influence on the structure of the resulting numerical method. Let us finally recall that the construction of numerical methods for the full problem involves also discretization of the space and velocity variables. The latter discretization in particular is a challenging problem for the Boltzmann equation due to the high-dimensionality of the collision operator [8,,6,] and the occurrence of the Bose Einstein condensation phenomenon in the degenerate quantum case [,3]. Here, however, we do not discuss further these issues. The rest of the paper is organized as follows. In Sect. we review some basic features of the quantum Boltzmann equation and its Euler limit. We emphasize in particular the differences between the classical and the quantum equilibrium states. Net in Sect. 3 we introduce the general form of the asymptotic-preserving eponential methods for the quantum Boltzmann equation. The properties of the method are then analyzed in Sect. 4. Several numerical eamples are reported in Sect. to show the AP property and the high-order accuracy of the schemes. We conclude the paper with some remarks in the last section. The Quantum Boltzmann Equation and Its Euler Limit The quantum Boltzmann equation was first formulated by Nordheim, Uehling and Uhlenbeck from the classical Boltzmann equation through heuristic arguments [6,3]. In its dimensionless form, the equation writes as: t f + v f = ε Q q( f ), t, R d, v R d, d =, 3, (.) where f (t,,v)is the phase space distribution function representing the (rescaled) number of particles that travel with velocity v at location and time t. ε is the so-called Knudsen number defined as the ratio of the mean free path over the typical length scale. It could vary across scales from ε O() to ε, depending on which, the system falls into the kinetic regime or fluid regime, respectively. The collision operator Q q models the interaction between quantum particles (here and in the rest of the paper, we always use the upper sign to denote the Bose gas and the lower sign to the Fermi gas): Q q ( f )= R d S d B(v v,σ)[ f f ( ± θ f )( ± θ f ) ff ( ± θ f )( ± θ f )]dσ dv, (.) where as usual, f, f, f,and f are short notations for f (t,,v), f (t,,v ), f (t,,v ), and f (t,,v ). (v, v ) and (v,v ) are the velocities before and after collision: v = v+v + v v σ, (.3) v = v+v v v σ, where σ is the unit vector along v v. Then the operator (.) can formally be split, in a self-evident way, into a gain and a loss term, Q q ( f ) = Q q + ( f ) f Q q ( f ). (.4) The collision kernel B is a nonnegative function that only depends on v v and cos θ (θ is the angle between σ and v v ). For variable hard sphere (VHS) particles, B is independent of scattering angle: 3

4 8 J Sci Comput () 6: 74 B = C γ v v γ, (.) where γ = corresponds to the Mawell molecules, and γ = is the hard sphere model. The parameter θ is some constant proportional to the Planck constant: θ = C h d. (.6) It characterizes the degree of degeneracy of the system in the sense that when θ, one recovers the collision operator for classical particles: Q c ( f ) = B(v v,σ)[ f f ff ]dσ dv. (.7) R d S d Compared with Q c, the quantum Boltzmann operator Q q involves more nonlinearity (it is cubic rather than quadratic). This new feature brings more compleities to both theoretical and numerical studies. We are particularly interested in the fluid regime, where macroscopic equations can be derived similarly as the classical case. To this aim, we first summarize the basic properties of Q q.. Q q conserves mass, momentum, and energy: R d R d Q q ( f ) dv = Q q ( f )v dv = R d Q q ( f ) v dv =. (.8) Then if one defines the macroscopic quantities: density, average velocity u, specific internal energy e, stress tensor P, and heat flu q as = f dv, u = v f dv, e = v u f dv, (.9) R d R d R d P = (v u) (v u) f dv, q = (v u) v u f dv, (.) R d R d the following local conservation laws can be obtained from Eq. (.) after multiplication by (,v, v /) T and integration w.r.t. v: t + (u) =, t (u) + (u u + P) =, (.) ( t e + u ) + (( e + u) u + Pu + q ) =.. Q q satisfies the Boltzmann s H-theorem: f ln ± θ f Q q( f ) dv. (.) R d Moreover, the equality holds iff Q q ( f ) = andiff f reaches the local equilibrium the quantum Mawellian (also called Bose Einstein or Fermi Dirac distribution): M q =. (.3) θ z e (v u) T ( ) Strictly speaking, θ = π d h m v N, wherem is the particle mass, and v are the typical values of length and velocity, N is the total number of particles. 3

5 J Sci Comput () 6: 74 9 The new macroscopic quantities z and T are the fugacity and temperature. They are related to and e via = (πt ) d θ Q d (z), (.4) e = d T Q d + (z) Q d (z), where Q ν (z) is the Bose Einstein/Fermi Dirac function of order ν [3]: Q ν (z) = ν Ɣ(ν) z e d, ( ) < z < forbose gas, (.) < z < for Fermi gas and Ɣ(ν) = ν e d is the Gamma function. Remark. Compared to the classical Mawellian: M c = e (v u) (πt ) d T, (.6) the quantum Mawellian M q is not a Gaussian function, and z, T depend nonlinearly on ande, the macroscopic quantities that could be readily obtained by taking the moments of f. In fact, it is not difficult to see that when z, Q ν (z) behaves like z. Therefore, in the system (.4), with and T fied, we send θ, and get z, e d T. (.7) (πt ) d θ Since z is very small, one can neglect in(.3), which results in M q z e (v u) T θ (πt ) d e (v u) T = M c. (.8) On the other hand, if θ is not small, M q and M c will be quite different from each other. Figure gives a simple illustration of the aforementioned two regimes, which we will refer to as (nearly) classical regime and quantum regime in the following discussion. The physical range of interest for a Bose gas is < z, where z = corresponds to the onset of Bose Einstein condensation (BEC). To avoid singularity, in this paper we do not consider this etreme case. We refer to [,3] for some recent results on the construction of numerical methods for the formation of BEC.. The Euler Limit Now as the Knudsen number ε ineq.(.), based on the discussion above, f is driven to the quantum Mawellian M q. Substituting M q into (.), we see that P = d ei and q = (I is the identity matri). Hence the system (.) can be closed and yields the following quantum Euler equations: t + (u) =, t (u) + (u u + d ei) =, (.9) t ( e + u ) + (( d+ d e + u) u ) =. 3

6 6 J Sci Comput () 6: 74 M c M q,b M c M q,f 4 (a) M c M q,b M c M q,f.... (b). Fig. Differences between M c and M q in a the classical regime; b the quantum regime. The left column is for Bose gas and the right one is for Fermi gas Obviously, written in terms of the macroscopic variables, u, and e, this system is eactly the same as classical Euler equations. The form would be much more complicated if everything is denoted in terms of z, u,andt. Remark. By performing a Chapman Enskog epansion to the net order, one can obtain the quantum Navier Stokes (NS) system which differs from its classical counterpart []. In particular, the viscosity and heat conductivity coefficients not only depend on e but also on. The design of a numerical scheme which is capable to capture with high accuracy the NS limit is actually under study and will be considered in future work. 3 The Asymptotic-Preserving (AP) Eponential Methods In this section we propose a class of high-order numerical methods for the quantum Boltzmann equation that gives accurate solution in both kinetic and fluid regimes, with time discretization not controlled by ε; is accurate for both classical and quantum regimes with accuracy analysis independent of θ (or Planck constant h). 3

7 J Sci Comput () 6: 74 6 We want to avoid computing the Eq. (.) directly: for numerical stability, if eplicit method is used, the time step size should be at most O(ε) which poses prohibitive numerical epenses in the fluid regime. On the other hand, implicit method is not feasible due to the complicated collision operator. We need an eplicit scheme that relaes h to be independent of ε but still captures the right asymptotic limit, i.e. to achieve AP property. To that end, we need to go through two steps: we first rewrite the Eq. (.) in an eponential form, and then apply the eplicit Runge Kutta methods on the newly derived equation. The difficulty is two fold: firstly, the equation needs to be reformulated in a way such that eplicit Runge Kutta, under very mild condition, automatically achieves AP property, and secondly, the new terms emerged in the new equation need to be treated with consistent schemes. We address these two difficulties in the following two subsections respectively. 3. Reformulation of the Equation and Basic Numerical Methods In this subsection, we reformulate the Eq. (.) in a form such that basic eplicit Runge Kutta methods automatically achieve asymptotic-preserving properties. The idea is adopted from [6,]: t [ ] [ ( f M q )e μt Qq ( f ) μ(m q f ) ε = ε v f t M q ] e μt ε. (3.) This equation is derived through simple calculation, and is completely equivalent to the original quantum Boltzmann Eq. (.). However, if eplicit methods are applied onto this equation instead of the original one, one could obtain the AP property. In the Eq. (3.), μ could be any constant independent of time, and M q is the local quantum Mawellian function. As it shares the same moments with f, we update them through the following equation: t φm q dv = t φ f dv = φ(v f )dv, φ = (,v, v /) T. (3.) In the derivation we also used the fact that the first d + moments of Q q are all zeros, as mentioned in (.8). Note that to insure the validity of (3.), μ could be any O() constant independent of time. If positivity is desired, we require: μ sup Qq, (3.3) v,t where Qq is the loss part of the collision operator (.4). We then apply the standard Runge Kutta method to Eqs. (3.) and(3.). The simplest eample is the forward Euler scheme: { ( f n+ Mq n+ )e λ = ( f n Mq n) + h ε φm n+ q dv = φ f n+ dv = φ f n dv h φ(v f n )dv, [ ] P n μmq n εv f n ε t Mq n, (3.4) where h is the time step, λ = μh/ε, and the operator P is defined as P( f ) = Q q ( f ) + μf. The more general κ-step eplicit Runge Kutta method gives: Stage i (i =,...,κ): ( f (i) M q (i) )e c i λ = ( f n Mq n) + i [ j= a ij h ε ] P ( j) μm q ( j) εv f ( j) ε t M q ( j) e c j λ, (i) φm q dv = φ f (i) dv = φ f n dv + i j= a ( ij h φv f ( j) dv ) (3.a) ; 3

8 6 J Sci Comput () 6: 74 Final Stage: ( f n+ Mq n+ )e λ = ( f n Mq n) + κ h i= b i ε [ ] P (i) μm q (i) εv f (i) ε t M q (i) e c i λ, φm n+ q dv = φ f n+ dv = φ f n dv + κ ( i= b i h φv f (i) dv ), (3.b) where f (i) stands for the estimation of f at time t = t n + c i h. a ij, b i,andc i are Runge Kutta coefficients that satisfy i j= a ij = c i and κ i= b i =. They are usually stored in a Butcher tableau as: c A b T (3.6) Clearly at each stage i, toevaluate f (i), one needs to find M q (i) at the new stage first, and a good approimation of t M q ( j) at old stages (this also applies to the final stage). Before moving to the net step some considerations are necessary. Remark 3. The above eponential approach applied to Eq. (3.) corresponds to the socalled integrating factor method [3]. Here we limit our analysis to this class of schemes, however, we refer to [,3,] for other possible eponential techniques that can be used to construct other types of AP eponential schemes. Reformulation (3.) holds true for arbitrary function M q which shares the same moments with f. We use the local Mawellian M q because this guarantees the strong AP property, as will be proved in Sect. 4. A simplifying assumption, analyzed in [], consists in taking M q constant along the time stepping so that the term t M q disappears and the scheme simplifies. This choice, although in general less accurate in intermediate regimes, permits to obtain AP schemes with better stability and monotonicity properties. We leave the analysis of this approach in the quantum case for future studies and refer to [] for further details. 3. Computation of M q (i) and t M q ( j) We now show how to evaluate M q (i) and t M q ( j) provided f ( j), M q ( j) ( j < i) are known from previous stages. Computation of M q (i). By definition in (.3), M q (i) is obtained once we have u (i), z (i),andt (i). The second equation in (3.a) gives the macroscopic quantities (i), u (i),ande (i), and thus to obtain z (i) and T (i), one only needs to invert the system (.4). Note that the system is nonlinear. In the implementation, we use the standard Newton-iteration. Details about the approimation and inversion of the quantum function Q ν (z) can be found in [4]. Computation of t M q ( j). This is the key idea of the scheme. Write M q = M q (z, T, u), it is not difficult to derive that (we drop the superscript ( j) for simplicity) t M q = M q ( ± θ M q ) [ z t z + (v u) T t T + v u ] t u. (3.7) T While t, t u,and t e can be directly obtained from the macroscopic equations as we shall see, the computation of t z and t T is, however, not eplicit. Therefore, it is desirable to 3

9 J Sci Comput () 6: transform the epression (3.7) in terms of t, t u,and t e. Through the straightforward but cumbersome calculations in the Appendi, we end up with: t M q = M q ( ± θ M q ) [A t + B t e + C t u], (3.8) where A = ) (v u) (M(z) + ( N(z)), (3.9) dt ( (v u) B = N(z) d ) et e M(z), (3.) C = v u T, (3.) and M(z) and N(z) are defined by: M(z) = Q d (z) Q ( d + ) Q d (z) d T 4e Q d (z), N(z) = d ( (z) d + ) Q d (z) d T 4e Q d (z). (3.) To compute t, t u,and t e,weuseeq.(3.) to transform the time derivative into spatial derivative, namely: t = t Mq dv = v f dv := F, t (u) = t vmq dv = v(v f )dv := F, ( t e + u ) = v t M qdv = (3.3) v (v f )dv := F 3, which is t = F, t u = ( F + F u), t e = ( F3 + F e + F u + u (F F u) ). (3.4) In this way, we could compute the time derivatives using only spatial discretizations, and the scheme is automatically consistent with the framework (3.). 4 Properties of the Eponential AP Methods In this section we briefly analyze the numerical scheme. We are going to show that our scheme is consistent, recovers the classical Boltzmann equation in the classical regime, and is AP. The choice (3.3)ofμ guarantees the positivity, and the proof was given in section 4. of []. We omit it here.. The classical regime: In the classical regime, θ (or the Planck constant) is considered as a very small number, and the fugacity z. In this regime, theoretically, the quantum Boltzmann equation recovers the classical Boltzmann equation, and our schemes should reflect this consistency. For z, as we have seen previously Q ν (z) z, ande d T. By definition, M(z), N(z). Plugging these relations back into 3

10 64 J Sci Comput () 6: 74 equation (3.8): we have A (v u), B d et e, C = v u T. (4.) Therefore, t M q M q [ t + ( d 4e (v u) d ) t e + d ] e e (v u) tu. (4.) This is indeed the time evolution of the classical Mawellian function t M c ; equation (.3): as argued in Remark., M q goes to M c ; equation (.): formally Q q also becomes the classical collision operator Q c as θ. Combining these three arguments, we see that the scheme (3.)becomes the Eponential AP method developed for the classical Boltzmann equation in []. We successfully recovers the classical regime.. Consistency: Here we assume the time step h resolves ε. We firstly rewrite the scheme (3.a)as: f (i) = M (i) q + j + ( f n M n q )e c i λ a ij h ε [ P ( j) μm q ( j) εv f ( j) ε t M q ( j) ] e (c j c i )λ. (4.3) As h is small, we Taylor epand the eponential term, and rewrite e c i λ c i λ and e (c j c i )λ + (c j c i )λ. We keep O() and O(h) terms and neglect higher orders, the scheme becomes: f (i) = + h + O(h ), (4.4) with = M q (i) + f n Mq n ; (4.) h = c i λ( f n Mq n ) + h [ ] a ij P ( j) μm q ( j) εv f ( j) ε t M q ( j). ε j (4.6) As M q (i) is the Mawellian obtained with macroscopic quantities evaluated at time t n +c i h, and thus the difference M q (i) Mq n is at most order h, therefore, one has On the other hand, we rewrite h as: h = ( ) Q ( j) a ij h v f ( j) ε j = f n + c i h t M n q + O(h ). (4.7) + j a ij λ( f ( j) M q ( j) ) c i λ( f n Mq n ) j a ij h t M ( j) q. 3

11 J Sci Comput () 6: 74 6 As f ( j) f n = O(h), M ( j) q h = j a ij h Combining Eqs. (4.7)and(4.8), we have f (i) = f n + h j Mq n = O(h), and j a ij = c i, we rewrite it as: ( Q ( j) ε v f ( j) ) c i h t M n q + O(h ). (4.8) ( ) Q ( j) a ij v f ( j) + O(h ), (4.9) ε and the consistency of the scheme is obvious. We could perform the same analysis to (3.b) and the proof will be omitted from here. 3. Asymptotic preserving: Here we show AP property of the numerical scheme, namely, as ε, the distribution function will automatically capture the solution to the Euler equation. For simplicity, we only show proof for the case when c < c < < c κ <. The argument presented here will no longer hold if any sub-stage share the same time step, i.e. c i = c i+ for some i, but we still have the same conclusion. The proof for that more general case could be found in []. We still use the formula (4.3). As c i monotonically increases, in the zero limit of ε, λ and the second and the third terms in (4.3) vanish, leaving: f (i) = M (i) q + O ( λe cλ) M (i) q, i =,...,κ, with c = min i c i+ c i >. We take the moment of both sides, and combine it with the second equation in the scheme (3.a): φ f (i) dv φ f n dv a ij h φv M q ( j) dv. (4.) j Similarly for the numerical solution from (3.b) we obtain φ f n+ dv φ f n dv b i h φv M q (i) dv. (4.) i This is eactly how we close the moment system and obtain the Euler equation analytically, and thus we capture the Euler limit. Let us note that the limiting resulting scheme (4.) (4.) is nothing but the underlying eplicit Runge Kutta method, used in the construction of the eponential scheme, applied to the limiting Euler system. Therefore the method is not only consistent but it preserves the order of accuracy in the fluid limit. Numerical Eamples In this section we present several numerical results using our schemes. The first eample is to show the AP property. The reference solution is given by directly applying the forward Euler method on (.). In this case, for stability reason, time step h has to be small, and controlled by ε. In comparison, h only needs to satisfy the CFL condition in our method. The second eample verifies the high-order accuracy. The value μ in all eamples is chosen as μ =.. Note that both eamples are performed for in D and v in D. Ep-RK is referred to as RK in time coupled with second-order La-Wendroff scheme with van Leer 3

12 66 J Sci Comput () 6: 74 limiter in space [9]. Ep-RK3 is referred to as RK3 in time coupled with standard WENO3 in space [3]. The Butcher tableau of RK (midpoint) and RK3 (Heun method []) we used in computation are given as follows: / / /3 /3 /3 /3 /4 3/4 (.) For the velocity discretization, we use 64 points in each direction and perform the fast spectral method [6] for Mawell molecule kernels. We use v ma = 7.86, which is determined by the criteria of the spectral method for the collision operator []. If our AP scheme is used, h time step is chosen to satisfy the CFL condition v ma.. Furthermore, functions M(z) and N(z) (in the evaluation of t M q ) have the following simple form when d = : where for M(z) = Q (z) Q (z) T e Q (z), N(z) = Q (z) Q (z) T e Q (z), (.) Bose gas: Q (z) = ln( z), Q (z) = z Fermi gas: Q (z) = ln( + z), Q (z) = z. Sod Problem z ; +z. In this subsection we compute a Sod problem. In this problem, in the limiting Euler regime, the solution should have a shock, a rarefaction and a contact discontinuity. The initial data for the macroscopic quantities are chosen as: { =, u =, u y =, T = ; (.3) =., u =, u y =, T =.. For the microscopic quantities, we choose f (t = ) to be a summation of two Gaussians, as shown in Fig., and thus is far away from the quantum equilibrium. Figures 3 and 4 show the numerical results using our new schemes. We consider both classical regime (θ =.) and quantum regime (θ = 9) (the behaviors of Bose gas and Fermi gas in the classical regime are very close to the classical gas, thus one of them is omitted). In the case when Knudsenε =. (kinetic regime), the reference solutions are given by directly applying the forward Euler scheme onto the original Boltzmann equation with the spacial discretization = /6, and time step h = /,6, and our method uses = /8 and h = /,8. When ε = 6 (fluid regime), for reference data, we could not afford the fine discretization any longer since an eplicit scheme would require h = O( 6 ), and thus we directly compute the limiting Euler equation. In contrast, our new scheme only uses = /6 and h = /,6 O( 4 ), much bigger than the Knudsen number ε. We also measured the difference between the distribution function f and the Mawellian M q.infig. we can clearly see that smaller ε gives faster convergence towards the Mawellian. For comparison, we mention that the same initial data (.3) was used in [8] for testing the first-order IMEX AP scheme. The results we obtained here are very similar to those but with better resolution (due to the higher-order accuracy of the scheme). 3

13 J Sci Comput () 6: F(t=) F(t=) M q (t=) Fig. Sod problem. The initial distribution at =. The figure on left is f (t = ) and right is f M q.8 ε=, θ=., t=., BE forward Euler Ep RK.8 ε=, θ=9, t=., FD forward Euler Ep RK.8 ε=, θ=9, t=., BE forward Euler Ep RK (a) e ε=, θ=., t=., BE forward Euler Ep RK e ε=, θ=9, t=., FD forward Euler Ep RK e ε=, θ=9, t=., BE forward Euler Ep RK (b) ε=, θ=., t=., BE forward Euler Ep RK ε=, θ=9, t=., FD forward Euler Ep RK.8.7 ε=, θ=9, t=., BE forward Euler Ep RK..6 z..8.6 z.. z (c) Fig. 3 Sod problem. ε = (kinetic regime). The three columns, from the left to the right are for Bose gas in classical regime, Fermi gas in quantum regime and Bose gas in quantum regime. The three rows present density, internal energy e and fugacity z. Note the scale differences for the three cases 3

14 68 J Sci Comput () 6: 74.8 ε= 6, θ=., t=., BE limiting Euler Ep RK.8 ε= 6, θ=9, t=., FD limiting Euler Ep RK.8 ε= 6, θ=9, t=., BE limiting Euler Ep RK (a) e ε= 6, θ=., t=., BE limiting Euler Ep RK e ε= 6, θ=9, t=., FD limiting Euler Ep RK e ε= 6, θ=9, t=., BE limiting Euler Ep RK (b) ε= 6, θ=., t=., BE limiting Euler Ep RK 3. 3 ε= 6, θ=9, t=., FD limiting Euler Ep RK.8.7 ε= 6, θ=9, t=., BE limiting Euler Ep RK z z z (c) Fig. 4 Sod problem. ε = 6 (fluid regime). The three columns, from the left to the right are for Bose gas in classical regime, Fermi gas in quantum regime and Bose gas in quantum regime. The three rows present density, internal energy e and fugacity z. Note the scale differences for the three cases 8 3 θ =, Bose gas 8 3 θ =, Fermi gas 7 7 ma(f Ma q ) ma(f Ma q ) ε=. ε=. ε= 3 ε= 3 ε= 6 ε= time time Fig. Sod problem. The difference between the distribution function f and the Mawellian M q decays in time. The figure on the left is for Bose gas and the right one is for Fermi gas. θ = 3

15 J Sci Comput () 6: Convergence Rate Test In the second eample we show the convergence rate. We use the following smooth initial data: = cos (π ); e = cos (π ); (.4) u = u y = ; h is chosen such that the CFL number is. (independent of ε). Note that this is the unique stability restriction that we must impose in our numerical discretization. To measure the convergence rate, we check the L error of and compute the decay rate using: i (t) i (t) Error i = ma. (.) t=t n i (t) Here the notation i is computed on i (with i being a integer) grid points. Theoretically, if a numerical scheme is of k-th order, then the error should decay as: Error i < C (i) k for h small enough. In each subfigure in Fig. 6, we show the convergence rate with θ = andθ = using Ep-RK and Ep-RK3. We perform the same test for both Bose gas and Fermi gas in both kinetic regime and fluid regime. The numerical results are in good agreement with our theoretical epectation..3 Miing Regime In the last eample, we show the numerical results for a problem with miing regime. This eample is adopted from []. We consider the initial data at equilibrium with macroscopic quantities: sin (π )+ = 3, cos (π )+3 4, T = u = u y =. (.6) We compute the Fermi gas with θ =.. The problem has miing regime, and ε varies across kinetic and fluid regimes: ε = ε +. tanh ( ) +. tanh ( + ) (.7) with ε = 3. Figure 7 shows the profile of the Knudsen number ε. This problem is difficult if the standard method is applied: the time step is controlled by the smallest Knudsen number although in most part of the domain, the problem is in kinetic regime and time step could have been larger. The AP method we presented here, on the other hand, does not rely on any time constraint beside the CFL condition. In Fig. 8 we show the numerical results given by our method. We used = /4, and h = /64, and this discretization is bigger than ε. The reference solution is computed through eplicit method with denser mesh: = /6, and the time discretization resolves ε : h = /,. We observe good agreement in all quantities. As the purpose here is show the effectiveness of the AP scheme, but not the solver for collision operator, to save the computational cost (especially for the eplicit method), the computation in velocity space was done using the quantum BGK model. 3

16 7 J Sci Comput () 6: ε =, Bose gas RK, planck = RK3, planck = RK, planck =. RK3, planck = ε = 6, Bose gas RK, planck = RK3, planck = RK, planck =. RK3, planck =. log(error ) log(error ) log(d) (a) log(d) 3 ε =, Fermi gas ε = 6, Fermi gas log(error ) RK, planck = RK3, planck = RK, planck =. RK3, planck = log(d) (b) log(error ) RK, planck = RK3, planck = 8 RK, planck =. RK3, planck = log(d) Fig. 6 Convergence rate test. The left column is for ε = andε = 6 on the right.thetop figures are for Bose gas and the bottom ones areforfermigas Fig. 7 Miing regime: the Knudsen number varies across regimes.. ε kinetic regime fluid regime

17 J Sci Comput () 6: t=. EpRK N8 EpRK3 N8 ref:n=6.3.. t=. EpRK N8 EpRK3 N8 ref:n= u e t=. EpRK N8 EpRK3 N8 ref:n=6 z t=. EpRK N8 EpRK3 N8 ref:n= Fig. 8 Miing regime. Top: density and velocity u ; bottom: internal energy e and fugacity z 6 Conclusions In this paper we have etended the numerical approach recently introduced in [] to the case of the quantum Boltzmann equation. In particular, we have shown how to derive high-order asymptotic preserving schemes which work uniformly with respect to the Planck constant. Numerical results for second and third order methods confirm the robustness and accuracy of the present method. We did not tackle the issue of the formation of the Bose Einstein condensate since this involves also a careful choice of the velocity discretization whereas here we concentrate our attention on the time discretization problem only. In our future research we will focus on these challenging aspects. Acknowledgments We would like to epress our gratitude to the NSF Grant RNMS-7444 (KI-Net), and CSCAMM, University of Maryland for holding the conference Quantum Systems: A Mathematical Journey from Few to Many Particles in May 3, during which this work was initiated. 7 Appendi: Derivation of t M q In this appendi, we give the details of the derivation of (3.8). Our goal is to represent t z and t T in Eq. (3.7) in terms of t and t e. 3

18 7 J Sci Comput () 6: 74 First, combining the two equations in system (.4)gives Therefore, we define a function F(z) such that Q d + d (z) ( ) d d = θ Q d. (7.) (z) 4πe d + y = F(z) = Q d + d (z) Q d d + (z), (7.) and a function G(y) such that z = G(y) = F (y). (7.3) Then we have G (y) = F (z) = Qd d + (z) ( d + ) Q d d (z)q d (z)q d (z) d d + Q d. (7.4) d + (z)q (z)q d + d (z) + For the Bose Einstein/Fermi Dirac function, one has the following nice property (see [3]) Using (7.) in(7.4), G (y) = F (z) = = zq ν (z) = Q ν (z). (7.) zqd d + (z) ( d + ) Q d d (z)q d (z)q d (z) d d + Q d (z)q d + d zq d + d + (z) [ Q d ( d (z) d + ) Q d (z)q d +(z) d Q d (z) From the second equation of (.4) we know d (z)q d + d (z) ]. (7.6) then Note that G (y) = ( ( ) d ) d z = G θ 4πe e Q d + (z) = dt Q d (z), (7.7) z ( ) d e dt ( d + ) Q d (z) d T 4e Q d (z). (7.8), T = θ d ( π Q d (z) ) d, (7.9) 3

19 J Sci Comput () 6: so we have and t z = G ( = t T = θ d d πd d Q t d (z) = T d Therefore, T d t T d z (v u) t z + T t T = ( ) d ) ( ) d ( d d θ θ 4πe 4π zqd (z) ( d + ) Q d (z) d T 4e Q d (z) e d d Q d (z) d Q t z + = θ d πd d (z) Q d (z) Q d (z) z t z = T d t t d ) te e d + ), (7.) ( t d e te d d Q t d (z) d Q d (z) d zq t z + d (z) Q d (z) ( ( d + ) Q d (z) d T 4e Q d (z) t d ) e te. (7.) Q d (z) ( d + ) Q d (z) d T 4e Q d (z) Q (v u) + dt (v u) d (z) t ( dt d + ) Q d (z) d T 4e Q d (z) Q d (z) (v u) = ( + d + ) Q d (z) d T 4e Q d (z) dt Q (v u) d (z) + ( et d + ) d Q d (z) d T 4e Q d (z) e ( t d ) e e t ( t d ) e t e Q d (z) ( d + ) Q d (z) d T Q d (z) ( d + ) Q d (z) d T 4e Q d (z) t 4e Q t e. (7.) d (z) Then if we define M(z) and N(z) as in (3.), (3.8) follows readily from the above equation. References. Arlotti, L., Lachowicz, M.: Euler and Navier Stokes limits of the Uehling Uhlenbeck quantum kinetic equations. J. Math. Phys. 38, (997). Bennoune, M., Lemou, M., Mieussens, L.: Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier Stokes asymptotics. J. Comput. Phys. 7, (8) 3. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press, Cambridge (99) 4. Degond, P., Jin, S., Mieussens, L.: A smooth transition model between kinetic and hydrodynamic equations. J. Comput. Phys. 9(), (). Dimarco, G., Pareschi, L.: Fluid solver independent hybrid methods for multiscale kinetic equations. SIAM J. Sci. Comput. 3(), () 6. Dimarco, G., Pareschi, L.: Eponential Runge Kutta methods for stiff kinetic equations. SIAM J. Numer. Anal. 49(), 7 77 () 3

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