c 2017 Society for Industrial and Applied Mathematics

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1 MULTISCALE MODEL. SIMUL. Vol. 5, No. 4, pp c 07 Society for Industrial and Applied Mathematics THE VLASOV POISSON FOKKER PLANCK SYSTEM WITH UNCERTAINTY AND A ONE-DIMENSIONAL ASYMPTOTIC PRESERVING METHOD YUHUA ZHU AND SHI JIN Abstract. We develop a stochastic asymptotic preserving s-ap scheme for the Vlasov Poisson Fokker Planck system in the high field regime with uncertainty based on the generalized polynomial chaos stochastic Galerkin framework gpc-sg. We first prove that, for a given electric field with uncertainty, the regularity of initial data in the random space is preserved by the analytical solution at a later time, which allows us to establish the spectral convergence of the gpc-sg method. We follow the framework developed in [S. Jin and L. Wang, Acta Math. Sci., 3 0, pp. 9 3] to numerically solve the resulting system in one space dimension and show formally that the fully discretized scheme is s-ap in the high field regime. Numerical examples are given to validate the accuracy and s-ap properties of the proposed method. Key words. Vlasov Poisson Fokker Planck system, uncertainty quantification, asymptotic preserving, polynomial chaos, stochastic Galerkin AMS subect classifications. 65M70, 8D0 DOI. 0.37/6M Introduction. In this paper we are interested in developing a stochastic asymptotic preserving scheme for the Vlasov Poisson Fokker Planck VPFP system with random inputs, which arises in the kinetic modeling of the Brownian motion of a large system of particles in a surrounding bath []. One application of such a system is in electrostatic plasma, in which one considers the interactions between the electrons and a surrounding bath via the Coulomb force. The equation takes the form of a Liouville equation with a Fokker Planck operator in the velocity space, coupled with a Poisson equation for the electric field. See section for details of the equations. The unknown in the system is ft, x, v, the particle density distribution of particles at time t > 0, position x with velocity v. In addition to the classical difficulty of high dimensionality to solve equations in the phase space, the problem under study has two more computational challenges: multiscale and uncertainty. In this paper the high field regime, in which the strong forcing term balances the Fokker Planck diffusion term [], will be considered. In this problem, numerical stiffness arises due to the strong field and diffusion term. On the other hand, in this regime one can approximate the VPFP system by its high field limit, which has the form of a transport-poisson system for the density and electric potential [0, 3]. One successful numerical strategy to efficiently compute into such asymptotic regimes is to develop asymptotic preserving AP schemes, which preserves the continuous Received by the editors August 8, 06; accepted for publication in revised form June 5, 07; published electronically October 4, Funding: This work was partially supported by NSF grants DMS-584 and DMS-079: RNMS KI-Net; by NSFC grant ; and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin Madison with funding from the Wisconsin Alumni Research Foundation. Department of Mathematics, University of Wisconsin Madison, Madison, WI yzhu3@wisc.edu. Department of Mathematics, University of Wisconsin Madison, Madison, WI 53706, and Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 0040, China sin@wisc.edu. 50

2 THE VPFP SYSTEM WITH UNCERTAINTY 503 asymptotic limit in the discrete space in a numerically uniformly stable way []. This strategy has been widely used in kinetic and hyperbolic equations with multiple time and space scales see [3] for a general review and [4] for applications in plasma. For its development for the high field limit, see [7, 8, 8, 3]. Another difficulty here is to treat the uncertainty. Due to modeling and measurement errors, uncertainties in kinetic modeling could arise from initial and boundary data and the forcing term. In this paper we will consider the cases in which the electric potential and initial data contain random inputs, modeled by random variables with given probability density functions. In recent years, the generalized polynomial chaos approximation based stochastic Galerkin gpc-sg methods have found many applications in a wide range of physical and engineering problems see [6, 6, 5], although its applications in kinetic problems are scarce see recent efforts in [9,, 5, 6, 4, ]. It is the goal of this paper to develop a gpc-sg method for the VPFP system with random inputs that are stochastic asymptotic preserving s-ap. As defined in [9], for the s-ap scheme, a stochastic Galerkin method for the VPFP system, in the high field limit, becomes a stochastic Galerkin method for the limiting transport-poisson system, when all the numerical parameters are held fixed. For this scheme, one can use a fixed mesh size, time step, and the number of gpc modes, in different asymptotic regimes. In particular, one does not need to numerically resolve the physically small scale and still capture the correct solutions of the high field limit. For a given electric potential that contains uncertainty thus the underlying problem becomes linear, we first prove, in section 3, that the system preserves the regularity of the initial data in the random space. In section 4 we introduce the gpc-sg method for the VPFP system, and the regularity result in section 3 naturally leads to the proof of the spectral accuracy of the method in section 5. Since the gpc- SG system is a vector version of the deterministic VPFP system, in section 6, in the one-dimensional case, we will use the AP scheme developed for its deterministic counterpart in [7] for time, spatial, and velocity discretizations, and the method is shown formally to be s-ap, namely, in the high field limit, it gives the gpc-sg method actually a kinetic scheme for the limiting system. Numerical experiments are conducted to demonstrate asymptotic property, accuracy, and other properties of the method in section 7. In the near future we will also develop multidimensional s-ap schemes for the VPFP system.. The background and model... The VPFP system with uncertainty. In the VPFP system with uncertainty, the time evolution equations of particle density distribution function ft, x, v, z under the action of an electrical potential φt, x, z are t f + v x f ɛ xφ v f = ɛ v [vf + v f], x φ = ρ h, t > 0, x, v, z I z, with the following initial condition: f0, x, v, z = f 0 x, v, z, x, v, z I z. Here the distribution function ft, x, v, z depends on time t, position x, velocity v, and random variable z I z R d. z is in a properly defined probability space

3 504 YUHUA ZHU AND SHI JIN Σ, A, P, whose event space is Σ and is equipped with σ-algebra A and probability measure P. φt, x, z is the self-consistent electrical potential, and hx, z is a given positive background charge with global neutrality relation 3 f 0 x, v, zdxdv = hx, zdx, and the density function ρt, x, z is defined as 4 ρt, x, z = ft, x, v, zdv. In addition, we define operators L, L φ as 5 6 Lf, φ = t f + v x f ɛ xφ v f ɛ v [vf + v f], L φ f, φ = x φ ρ h... The high field limit. Here we will show the formal limit of when ɛ 0. First, integrate over v, 7 t fdv + x vfdv R v x φfdv = v vf + v fdv. N R ɛ N R ɛ N Define the flux 8 After integrating by parts, one has = vfdv. 9 t ρ + x = 0. Then multiply v to both sides of and integrate over v, 0 ɛ t vf + ɛ vv x f = v v f x φ + v v [vf + v f]. Letting ɛ 0, it becomes which implies 0 = v [ v f x φ + vf + v f] dv, Therefore, one has 0 = f x φ + vf + v fdv. 3 = ρ x φ. Finally plugging 3 into 9, one gets the high field limit of system, { t ρ x ρ x φ = 0, 4 x φ = ρ h. For each fixed z, the rigorous proof for the high field limit of the VPFP system in one dimension can be found in [0, 3].

4 THE VPFP SYSTEM WITH UNCERTAINTY Regularity of the solution in the random space. In this section, we study the regularity of ft, x, v, z for a given potential function φt, x, z. In this setting, the equation is linear. This regularity will be needed to prove the spectral convergence of the gpc approximation in section 5. To simplify the notation we also assume z I z R. All the theory can be extended to z R d easily. Before we start, let us first define πz : I z R + as the probability density function of the random variable zω, ω Σ. So one can define a corresponding L π space with inner product, 5 < f, g > π := fgπz dz, I z and weighted norm in x, v, z space 6 f π = I z f πzdxdvdz 3.. Regularity of solution in the random space.. Theorem 3.. Given φt, x, z, if there exists some integer m > 0, and positive constants C f, C φ, such that zf l 0 π C f, z l x φ L C φ, for l = 0,..., m, then 7 l z ft π D l e G l t ɛ for l = 0,..., m, where D l = a l C f l!, a = max{c φ, }, G l = l +, Proof. For notational simplicity, we take N =. However, the proof can be easily extended to multidimensional x and v. First, multiply fπz to and integrate it over x, v, and z; after integration by parts, one gets 8 ɛ t f π = f π v f π. For l =,..., m, taking the lth derivative in z to, one gets 9 l ɛ t zf l + ɛv x zf l x φ v zf l l i z l i x φ v zf i = v v l z f + v zf l. Multiplying πz l zf and integrating over x, v, and z, the second and third terms vanish, so one has, for l =,..., m, 0 ɛ t l zft π = I z l + l z f π v l zf π. l i z l i x φ v zf i zfπz l dxdvdz Using Young s inequality and the boundedness of l z x φ, one gets l ɛ t left zftright l π Cφ l i v zf i π + l + zft l π v zf l π.

5 506 YUHUA ZHU AND SHI JIN Multiplying a constant A m l to and summing l from to m, then adding A m 0 8 gives m m l m ɛ t A m l zft l π Cφ A m l l i v zf i π + l + A m l zft l π = l= m + m A m l v zf l π m l=i+ C φ l i A m l A m i m l + A m l zft l π. v i zf π A m m v i zf π Letting A m m = and m l=i+ C φ l i A m l A m i = 0, for i = 0,..., m, becomes m m 3 ɛ t A m l zft l π l + A m l zft l π, and one has a linear system for A m i 4 C φ, i = 0,..., m : 0 0 m 0 C φ m.... Cφ. m m C φ A m 0 A m. A m m A m m = m 0 m. m m m m. Lemma 3.. Solving the linear system 4, one has 5 m! 0 < A m l b m l, where b = max{, C l! φ}. 6 Proof. See Appendix A for the proof. Therefore, by Lemma 3., and apply Gronwall s inequality to 3, one obtains m m m A m l zft l π e m+t ɛ A m l zf0 l π e m+t m! ɛ Cf b m l l! [ ] e m+t m ɛ Cf m! b 0! m + b l 4 l 7 3 bm m! e m+t ɛ C f, l=

6 THE VPFP SYSTEM WITH UNCERTAINTY 507 which implies 7 m z ft π a m m!e m+t ɛ C f, where a = max{c φ, } 3.. Regularity of v f in the random space. Theorem 3.3. Given φt, x, z, if there exists some integer m > 0, and positive constants C f, C φ, such that z l v f0 π C f, z l x f0 π C f, z l x φ L C φ, z l xφ L C φ, for l = 0,..., m, then 8 l z v ft π C l e L l ɛ t for l = 0,..., m, where C l = 3a l C f l!, a = max{c φ, }, L l = ɛ + C φ l. Proof. Applying l z v and l z x to, l =,..., m, gives 9 30 ɛ t l z v f + ɛv x l z v f + ɛ l z x f l l i z l i x φ z i vf x φ v z l v f = v zf l + v z l v f + z l vf; l ɛ t z l x f + ɛv x z l x f xφ v zf l x φ v x zf l l i x z l i φ v zf i l l i x z l i φ v z i x f = v v z l x f + v z l x f. Multiplying πz z l v f to 9 and πz z l x f to 30, and integrating over x, v, and z, one has, respectively, ɛ t z l v f π + ɛ < z l x f, z l v f > π l < l i z l i x φ z i vf, z l v f > π dxdv = 3 z l v f π z l vf π and ɛ t z l x f π By Young s inequality, one gets l l < l i x z l i φ z i v f, z l x f > π dxdv < l i x z l i φ z i v x f, z l x f > π < xφ l z v f, l z x f > π dxdv = l z x f π l z v x f π. 3 ɛ t z l v f π ɛ z l x f π + ɛ l z l v f π l + Cφ l i z i vf π z l vf π;

7 508 YUHUA ZHU AND SHI JIN 3 ɛ t l z x f π C φ + + l l z x f π + C φ l z v f π l + Cφ l i z i v f π l + Cφ l i z i v x f π z l v x f π. Summing the two inequalities yields ɛ t l z v f π + z l x f π ɛ + Cφ l z l v f π + z l x f 33 π Similarly, for l = 0, one has 34 l + Cφ l i z i v f π l + Cφ l i z i vf π z l vf π l + Cφ l i z i v x f π z l v x f π. ɛ t x f π + v f π C φ ɛ x f π + v f π x v f π v v f π. Multiplying A m l to 3 and summing it from to m over l, then adding A m 0 34, gives 35 ɛ t m A m l l z v f π + l z x f π m ɛ + C φ l A m l m + + m Letting A m m = and A m i 36 ɛ t m A m l Cφ l=i+ Cφ l=i+ m l i A m l z i v f π m l i A m l A m i l z v f π + l z x f π i z vf π + i z v x f π. solve 4, for i = 0,..., m, one has l z v f π + l z x f π m ɛ + C φ l A m l A m i z i v f π m + m ɛ + C φ l A m l l z v f π + l z x f π l z v f π + z l x f π ;

8 THE VPFP SYSTEM WITH UNCERTAINTY 509 then by Lemma 3. and Gronwall s inequality, one obtains 37 m A m l l z v f π + l z x f π 7 3 bm m! e ɛ+cφ+5+m ɛ t C f. Therefore, one can get 38 m z v f π 3a m m!e ɛ+c φ +5+m ɛ t C f. which completes the proof. Remark 3.4. Theorems 3. and 3.3 imply that if f and v f are in H m = {f l zf π <, 0 l m} initially, then under suitable assumption on the regularity of φ as given in Theorems 3. and 3.3, f, v f remain in H m at a later time. Thus the regularity in z of the initial data is preserved in time. However, the estimates are not sharp. For the linear transport equation with the special case of an isotropic collision kernel, a sharp uniform spectral convergence was established in [4], while for the anisotropic collision kernel, uniform in ɛ regularity was established in [5]. Uniform in ɛ regularity for the general linear uncertain kinetic equation with one conservation law was obtained in [] for both high field and parabolic limits. Obtaining sharp estimates for the VPFP system remains the subect of a future work. 4. The gpc method for the VPFP system. 4.. The method of gpc. Let W K π be the orthogonal polynomial space corresponding to the random space Σ, A, P, 39 W K π = {g : I z R : g span{φ k z} K }, where Φ k, k = 0,..., K, is a set of d-variate orthonormal polynomials of degree k satisfying 40 < Φ k, Φ l > π = EΦ k Φ l = Φ k zφ l zπzdz = δ kl. I z Here E means the expected value, and δ kl is the Kronecker delta function. By the classical approximation theory, W π is a Hilbert space with inner product <, > π. Thus the solution ft, x, v, z, φt, x, z to can be represented as 4 ft, x, v, z = f k t, x, vφ k z, φt, x, z = φ k t, xφ k z in L π. In the gpc-sg method, one seeks an approximation to the exact solution f and φ in the subspace W K π, i.e., the approximation solution ˆf K, ˆφ K are in the form of 4 43 K ˆf K t, x, v, z = ˆf k t, x, vφ k z ˆf K Φ K, K ˆφ K t, x, z = ˆφ k t, xφ k z ˆφ K Φ K,

9 50 YUHUA ZHU AND SHI JIN where φ K = Φ 0,..., Φ K, and ˆf k =< ˆf K, Φ k > π, ˆφk =< ˆφ K, Φ k > π, which are independent of z, are the components of vector ˆf K, ˆφ K satisfying, for 0 K, < L ˆf K, ˆφ K, Φ > π = 0, 44 < L φ ˆf K, ˆφ K, Φ > π = 0. We also approximate the given charge h by 45 K ĥ K x, z = ĥ k Φ k ĥk Φ K, where ĥkx =< h, Φ k > π, for k = 0,..., K. By the definition of ρ in 4, the numerical approximation of ρ is 46 K ˆρ K t, x, z = ˆρ k Φ k ˆρ K Φ K, where ˆρ k t, x = ˆf k t, x, vdv for k = 0,..., K. By 44, we have for each = 0,..., K, 47 { t f + v x f K ɛ k, xφ k v f l E kl = ɛ v [vf + v f ], x φ = ρ h for K, where E, 0 K, is a K + -dimensional matrix, and E k l = EΦ Φ l Φ k. In order to express the system in a simple form, and for the sake of combining the stiff terms and forming an AP scheme as in [7], we give the following lemma. Lemma 4.. For matrix E i, 0 i K, defined above, one has [ K K ] 48 x φ k v f l E kl = v E kˆf K x φ k. k, Proof. 49 K x φ k v f l E kl k, = K N K N xi φ k vi f l E kl = xi φ k k, K l= E k l vi f l [ K N K ] K N = xi φ k vi E k l f l = xi φ k vi Ekˆf K = K v [ x φ Ekˆf K,..., x N φ k Ekˆf K [ K ] = v E kˆf K x φ k. k= ]

10 THE VPFP SYSTEM WITH UNCERTAINTY 5 Now by Lemma 4., 47 can be written in a vector form as [ tˆf K + xˆf K v K ] ɛ v E kˆf K x φ k = ɛ v [ˆf ] K v K + vˆf, 50 ˆφK x = ˆρ K ĥk. 5. The spectral convergence of the gpc-sg method. In this section, we establish the spectral convergence of the gpc-sg method for a given potential φt, x, z. 5.. Stability. We first prove a stability result, estimating the evolution of ˆf K t π Theorem 5.. For all t > 0, 5 ˆf K t π e 3Nt ɛ ˆf K 0 π. Proof. Due to the orthogonality of φ k z, one has ˆf K π = ˆf K L, with L defined as L = 5 dxdv, R 3 where is the regular Euclidean norm for vectors. Therefore one only needs to prove the theorem for ˆf K t L. Multiplying ˆf to 47 and integrating over x and v, 53 = ɛ t ˆf + v x ˆf ɛ v v ˆf K k,l, xi φ k ˆf vi ˆfl E kl dxdv dxdv + N ɛ ˆf L ɛ v ˆf L. After integration by parts the second term on the left-hand side LHS vanishes, and the first term of the RHS becomes N ɛ R ˆf N dxdv. Summing from to K, one gets 54 t ˆf K L ɛ K k,l,i,=0 Note the second term on the LHS also vanishes, since 55 ɛ = ɛ K k,l,i,=0 N xi φ k ˆf vi ˆfl E kl dxdv ɛ + N ˆf K L ɛ. xi φ k ˆf vi ˆfl E kl dxdv K k, =0 + K xi φ k E k vi ˆf K k, l K xi φ k ˆf vi ˆfl E k l dxdv

11 5 YUHUA ZHU AND SHI JIN = ɛ + K k, >l K k, =0 K xi φ k E k vi ˆf K xi φ k E k l vi ˆf ˆfl dxdv, where the last inequality uses the symmetry of E k. Both terms in 5 vanish after integration by parts, so 54 implies N t ˆf K L ɛ + N 56 ˆf K L ɛ. By Gronwall s inequality, 57 ˆf K t L e 3Nt ɛ ˆf K 0 L, which completes the proof. 5.. The spectral convergence. Before we start to prove the convergence of the numerical approximation ˆf, for the sake of convenience, we assume z R, and all the proof can be easily extended to multidimensional z. We define operators L f, K as 58 L f := ɛ t + ɛv x v v N v, K := x φ v, then L = L f K. Let the proection of the exact solution ft, x, v, z to the subspace W K π be P K f, 59 K K P K f := < f, Φ k > π Φ k z := f k t, x, vφ k z := f K Φ K, where f = f 0,..., f K. As defined in 43, the numerical approximation ˆf K = ˆf K Φ K ; then the error can be split into two parts, f ˆf K = f P K f 60 + P K f ˆf K := R K + µ K, where 6 R K = is the proection error. Define vector k=k+ f k t, x, vφ k z 6 µ K = µ 0,..., µ K with µ i = f i ˆf i, i = 0,..., K. So µ K = µ K Φ K is the error of the gpc-sg approximation. Theorem 5.. Given φt, x, z, if for some integer m > 0, and positive constants C f, C φ, such that l z v f0 π C f, l z x φ L C φ, l z xφ L C φ, for l = 0,..., m, then for 0 < t < T, 63 µ K t π H me Lm+3N ɛ t K m, where H m = C AC mc φ Lm, with C A a constant depending on polynomials {Φ k z 0 k m}.

12 THE VPFP SYSTEM WITH UNCERTAINTY 53 Proof. Subtracting < Lf, Φ K > π = 0 by < L ˆf K, Φ K > π = 0, one has 64 < L f f ˆf K, Φ K > π < Kf ˆf K, Φ K > π = 0. Since L f is independent of z, 65 < L f f ˆf K, Φ K > π =L f < f ˆf K, Φ K > π = L f µ K. Plugging 65 into 64 gives 66 L f µ K < Kµ K + R K, Φ K > π = 0. Taking the dot product of µ K to 66, then integrating over x, v, yields [ 67 0 = Lf µ K µ K < Kµ K + R K, φ K > π µ K] dxdv = ɛ t µ K π N µ K π + v µ K π < KR K, µ K > π dxdv R N x φ v µ K πz dzdxdv I z = ɛ t µ K π N µ K π + v µ K π < KR K, µ K > π dxdv. This gives 68 ɛ t µ K π N + N µ K π + C φ v R K π. Since µ K 0 π = µ K 0 dxdv = 0, and by Grownwall s inequality, this implies 69 µ K t π ɛ C φ t 0 v R K s πds e 3N ɛ t. By the classical approximation theory and Theorem 3.3, 70 v R K π C A m z v f π K m C Lm AC m e ɛ t K m, where C A is a constant depending on polynomials {Φ k z 0 k m}. Plugging 70 into 69 yields 7 µ K t π H me Lm t ɛ e 3N ɛ t, K m where H m = C AC mc φ Lm, which implies 7 µ K t π H me Lm+3N ɛ t K m.

13 54 YUHUA ZHU AND SHI JIN Theorem 5.3. Assume φt, x, z, if for some integer m > 0, and positive constants C f, C φ, such that zf0 l π C f, z l v f0 C f, z l x φ L C φ, z l xφ L C φ, for l = 0,..., m. Then the Kth order numerical approximation ˆf K converges to the solution f with an error, 73 where O m = C A D m e Gm ɛ C φ, and ɛ. Proof. f ˆf K π O m K m, t +H m e Lm+3N ɛ t is a finite positive constant depending on C f, 74 f ˆf K π R K π + µ K π C A m z f π K m C AD m e Gm ɛ t + H m e Lm+3N ɛ t K m. + H me Lm+3N ɛ t The first inequality is because of the definition in 60, the second inequality is because of the error for proection and Theorem 5., the third inequality is because of Theorem 3.. Remark 5.4. Theorem 5.3 shows that for smaller ɛ, one needs larger K to get good accuracy. This motivates the development of the s-ap scheme in which one can take K independent of ɛ. As mentioned earlier, our regularity established in section 3 is not sharp; therefore, the convergence rate in Theorem 5.3 is not optimal. However, even if one obtains sharp estimates, the time and spatial discretizations still need to be AP. This is what the subsequent sections will be aimed at. 6. The s-ap schemes. 6.. The high field limit of the gpc method. We will first formally derive the high field limit of the gpc system 50. Integrating 50, and letting ĵk = R ˆf K v dv be the flux, one gets K m 75 t ˆρ K + x ĵk = 0; then, multiplying v, the transpose of v, to 78 and integrating it over v gives 76 K E k ˆρ K x φ k + ĵk = 0. Plugging 76 into 75 yields the high field limit system for the coefficient of ˆρ K and ˆφ K, K t ˆρ K x E k ˆρ K x ˆφk = 0, 77 ˆφK x = ˆρ K ĥk. This system is exactly the gpc system for the high field limit with uncertainty 4, which shows that the gpc system is AP.

14 THE VPFP SYSTEM WITH UNCERTAINTY The fully discrete first order scheme. Here we ll give the VPFP system with uncertainty a fully discrete scheme when N =. First we combine the stiff terms v [ K ˆφ x k E kˆf K ] and v vˆf K + vˆf K ; then [ tˆf K + v xˆf K = K ] ɛ v x ˆφk E k + vi K ˆf K K + vˆf, 78 xx φ K = ˆρ K ĥk, where I K is a K K identity matrix. Here we denote 79 K F = x ˆφk E k, P = F + vi K, A = P, where 80 P := P P. Let 8 M = π e A. Concerning the properties of the matrix M, we give the following proposition. Proposition 6.. Suppose M is defined in 8; then a v M = P M; b M is invertible, and M = πe A, v M = P M ; c M and M are both symmetric and positive definite; d Mv Mv is symmetric and positive definite for any v, v and Mv Mv = Mv Mv ; e R Mdv = I K, R vmdv = F ; f MP M = P. Proof. See Appendix B for the proof. Back to system 78, where the stiff terms can be represented by v [M v M ˆf K ] from Proposition 6.a, b, g, and thus 78 is equivalent to 8 tˆf K + v xˆf K = ɛ v [M v M ˆf ] K, xx φ K = ˆρ K ĥk. Denote ˆf i n = ˆft n, x i, v, 0 i N x, Nv Nv, n 0. N x, N v even are numbers of mesh points in the x and v directions, respectively. Let x i = a + iδ x, v = δ v, ˆρ n Nv/ i = δ ˆf n v = N v/ i, be the numerical approximation of density ˆρ. We choose N v sufficiently large such that outside the velocity domain, 83 f v Nv δ v 0, M v Nv δ v 0, during the computational time.

15 56 YUHUA ZHU AND SHI JIN We basically adopt the scheme in [7] for the deterministic problem. The first order scheme is ˆf n+ i ˆf n i δ t + n ˆf i+, ˆf n i, = ˆf δ x ɛ P n+ i, ˆφn+ x i = ˆρ n+ i ĥn+ i, where the upwind flux is used for spatial discretization, 86 n ˆf i+, = v + v n ˆf i, + v v n ˆf i+,. P ˆf n+ i is the discretization form of Pˆf = v [M v M ˆf], which is defined as P ˆf = δ v [ M +/ [ v M ˆf]+/ M / [ v M ˆf] / ] = [ δv M / + M / M+ˆf + M ˆf M / M / M ˆf M ˆf ] 87 = M / δv [ M /ˆf + + M / + + M / M / M / ˆf + M /ˆf ]. The algorithm is implemented as following: Step. Sum 84 over. Since the RHS vanishes, one gets 88 where F n i+ ˆρ n+ i ˆρ n i + δ t F n i+ = δ v f n i+,. This gives ˆρn+ i. F n i δ x = 0, Step. By using a Poisson solver, one gets gives M n+ i as 89 ˆφ n+ i from 85, which in turn M n+ i, = exp K ˆφ k n+ i+ ˆφ k n+ i E k + v I K. π δ x Step 3. Since Fi n = K ˆφ k n i+ ˆφ k n i E δx k can be decomposed as Fi n = Q n i Λn i Qn i, where Q n i is an orthogonal matrix, Λ n i = diagλ 0,..., λ K n i is a diagonal matrix. Then Mi n = Qn i e v+λn i Q n i. Therefore, letting Λ n i = e 4 v+λn i, 87 can be written as 90 P { n+ QΛ [ ˆf i = ɛ v Λ + Q ˆf+ Λ + + Λ Λ +Λ ] } n+ Q ˆf. i Λ Q ˆf

16 THE VPFP SYSTEM WITH UNCERTAINTY 57 Multiply Λ n+ i Q n+ i to 84, and let ĝ n+ i = Λ n+ i Q n+ one has [ ] Λ ĝ n+ i,+ n+ i,+ + Λ i, Λ i + ɛ v ĝ n+ i + ĝ n+ i, t 9 = ɛδ v Λ n+ i Q n+ i n ˆf i+, ˆf n i, δ x ˆf n i δ t. n+ i ˆf i ; Let b n i = Λ n+ i Q n+ i n ˆf i+, ˆf n i, δ x ˆf n i δ t ; then one has a scalar solver for each component ĝ n+ k of ĝ n+, k = 0,..., K, 9 [ mk g k n+ i,+ n+ i,+ m k n+ i + m ] k n+ i, m k n+ + e v g k n+ i + g k n+ i, t = b k n i, i n+ where m k n+ i = e v +λ k i 4, where it has been proved in [0] that the linear system for g k n+ i is positive definite, so one can invert it by the conugate gradient method. Remark 6.. Instead of using M + = M M +, one can also use M +/ = M ++M. By setting g i, = Λ i, Q i f i,, for fixed i, n, 87 will become P ˆf = [ M+ + M 93 δv M +ˆf + M ˆf M + M ] M ˆf M ˆf = Q [ Λ δv + + Λ ĝ+ Λ + + Λ + Λ ĝ + Λ + Λ ] ĝ. Thus, 9 becomes [ Λ + + Λ ĝ+ Λ δv Λ + Λ ĝ + Λ + Λ ] n+ ĝ i 94 = δ vq ˆf n i δ t + ˆf n i+, ˆf n i, δ x which can be decomposed to a scalar solver for each component of ĝ n+. In addition, it is easy to see the coefficient in 94 is a diagonally dominated matrix with negative diagonal entries, so it is a negative definite matrix The s-ap property Mass conservation. Since Pˆf has the property of mass conservation, its discretization P ˆf should have the same property. Let 95 K = M / + M / M+ˆf + M ˆf ;,

17 58 YUHUA ZHU AND SHI JIN then, by 87, 96 P ˆf = δ [K K ] = v δv K δ v = δv K δv K = 0. K Thus, summing 84, one can get the scheme for ˆρ n+, 88, which also implies i ˆρn+ i = i ˆρn i The formal proof of s-ap. Here we want to prove the scheme is s-ap, that is, for fixed δ t, δ x, δ v, when ɛ 0, it automatically becomes a gpc-sg approximation for the high field limit. Lemma 6.3. In scheme 84, ˆf i n Miĉn n i, as ɛ 0, where ĉn i of. 97 is independent Proof. For fixed i, n, let ɛ 0, multiply v to 84, and sum it over ; one gets 0 = v P ˆf = δv v [K K ] = δv δ v K, which is equivalent to 98 K = 0. Letting ɛ 0, 84 also implies P ˆf = 0 for all, or equivalently, 99 K K = 0. δ v This implies 00 K = c, where c is a constant depending on i and n. From 98, 00, one has K 0. By the definition of K in 95, this implies 0 M n i,+ ˆf n i,+ M n i ˆf n i = 0; therefore, 0 M n i ˆf n i = ĉ n i, and this gives ˆf n i = M n iĉn i. Lemma 6.4. If ˆf n i = M n iĉn i, where ĉn i is a constant vector, then ĉ n i = ˆρn i +Oδ v.

18 03 Proof. As defined in section 6., THE VPFP SYSTEM WITH UNCERTAINTY 59 δ v = Nv = Nv ˆfi = ˆρ i = = Nv δ v = Nv M i ĉ i. Since for fixed i, n, M i = π exp Fi+vI, where F is a constant symmetric matrix for each i. So there exists a unity matrix Q, and a diagonal matrix Λ = diagλ,..., λ K, s.t. F = Q ΛQ. Thus, 04 M = Q e Λ +v Λ+v I Q = Q diag π π e λ +v,..., e λn+v Q. Using the trapezoidal rule and assumption 83, 05 = R π e λ i +v dv = + exp π + exp π = Nv = Nv + λ i + v Nv e λ i +v π λ i + v Nv + Oδ v. Again by assumption 83, π exp λ i + v Nv + π exp λ i + v Nv Oδv, so 0 implies so δ v = Nv = Nv M = Q diag = Nv = Nv = Nv δ v = Nv e λ i +v + Oδv, π = e λ +v,..., δ v π Nv = Nv e λn+v Q π 06 Therefore, 07 = Q + Oδ viq = + Oδ v I. δ v M = + Oδ v I = + Oδ v I. So by 03 and 07, one gets ĉ n i = ˆρn i + Oδ v. Theorem 6.5. The first order scheme defined as is s-ap. That is, when ɛ 0, the limit of the first order scheme coincides with the gpc-sg discretization of high field limit 4.

19 50 YUHUA ZHU AND SHI JIN Proof. From Lemmas 6.4 and 6.3, as ɛ 0, 08 n ˆf i Mi n ˆρ n i + Oδv. Thus, F + = 09 = = R 0 F v + v Mvdv = 0 vi K + F e A dv π e A da F π F vmvdv 0 F Mvdv P e dp = F e F erff, π π where erfx = x π e t dt, F, P, A is defined in 79. Similarly, F v v 0 = Mvdv = vmvdv = F 0 e F erf F. π Then F n i+ R defined in 88 becomes F n i+ = F + ˆρ n i + F ˆρ n i+ + Oδ v, which is exactly the numerical flux of the kinetic scheme for 77 by [7, Chapter 3]. So as ɛ 0 88 becomes the forward Euler in time and kinetic scheme in space for the resulting system of the high field limit equation with uncertainty 4, which completes the proof for the s-ap property A second order scheme. Using the backward difference formula for time discretization [9] and the MUSCL scheme for space discretization, the second order scheme is given by 3 Here, 3ˆf n+ i 4ˆf i n n + ˆf i n n + v xˆf δ i v xˆf i = ˆf t ɛ P n+ i ˆφn+ x i = ˆρ n+ i ĥn+ i by Poisson solver. 4 and 5 v xˆfi = v ˆfi+, ˆf i, δ x ˆfi+, = ˆf i, + ψ θ + i+ ˆfi+, = ˆf i+, ψ θ i+ ˆf i+ ˆf i, v > 0, ˆf i+ ˆf i, v < 0, where θ + = ˆf i ˆf i and θ = ˆf i+ ˆf i+ are smooth indicators, and ψ = max0, i+ ˆfi+ ˆf i i+ ˆfi+ ˆf i min, θ is the slope limiter function []. The AP property can be similarly established as the first order scheme, so we omit the details here.

20 THE VPFP SYSTEM WITH UNCERTAINTY 5 7. Numerical examples. We solve the one-dimensional VPFP system with uncertainty, t f + v x f ɛ xφ v f = ɛ v[vf + v f], 6 + λ z xx φ = ρ h, x [x 0, x I ], v R, with periodic function φt, x, z satisfying 7 φt, x 0, z = φt, x I, z = 0, and only in section 7.3., λ 0. Initial conditions are given by 8 ρ 0 = ρ 0 x, λ z, f 0 = f 0 x, v, λ z, and the given positive charged background hx, z satisfies the global neutrality relation. Here z = z, z are two independent random variables following the uniform distribution U[a, b]. Given the gpc coefficients ˆf m, m = 0,,..., K of the numerical approximation ˆf K, the statistical quantities such as expectation and standard deviation are retrieved as 9 E[ ˆf K ] = ˆf 0, S[ ˆf K ] = K ˆf m. 7.. The order of convergence. This section is devoted to check the spectral convergence. The initial data is given by an C function in z U[0, ] and periodic in x: 0 m= ρ 0 x, z = + sinxe z, f 0 = ρ 0x, z e v+ xφx,z, x 0, π. π In order to satisfy the global neutrality relation for the background charge h, i.e., equation 3, we set h 0 x = + sinxz, periodic in x 0, π. Define the l -error for the expectation and standard deviation of the approximation solution ˆf K, error E = δ x δ v Ef i E ˆf i K, error S = δ x δ v Sf i S ˆf i K, i, where f, the reference solution, is calculated by the stochastic collocation method [4] with 0 Legendre quadrature points and mesh size δ x = π 000, δ t = δx 5, δ v = 400, while ˆf K is the numerical solution by the Kth order gpc-sg and the same mesh size as the reference solution. Figure is the l -error in terms of gpc order K for ɛ =, 0 3, 0 5, respectively, with fixed δ x, δ v, and δt. It shows exponential decay in K until the errors due to spatial, temporal, and velocity discretizations dominate. Furthermore, the amplitudes of the errors increase as ɛ decreases but are within the estimated numerical approximation errors. i,

21 5 YUHUA ZHU AND SHI JIN Fig.. Example 7.: Error of the numerical solution at T = 0.0 defined in when ɛ =, 0 3, 0 5. We take δ x = π δx, v [ 6, 6], δv =, δt = , 0 K The asymptotic preserving property. This section is devoted to checking the AP property of the scheme. We take the equilibrium initial data and nonequilibrium initial data, respectively. The certain part of the initial data in this example is the same as in section 3. in [7]. π 3 ρ 0 x, v, z = + cosπx + λ z, 4 hx, z = ecosπx + 0.z, x [0, ]. For the equilibrium initial condition, f 0 is given by 5 f 0 x, v, z = ρ 0x, z e v+ xφ, periodic in x [0, ], π while for the nonequilibrium initial data, f 0 is given by 6 e v+.5 + e v.5 f 0 x, v, z = ρ 0x, z π, periodic in x [0, ]. We study the evolution of the difference between f and equilibrium, 7 M eq = ρ π e v+ xφ, with respect to different ɛ as shown in Figure. Here the difference is defined as 8 difference = Ef EM eq = δ x δ v Ef i EM eq i. Figure shows the time evolution of the difference defined in 8 with different ɛ. One can see that whether the initial data is equilibrium or nonequilibrium, the s-ap method will push f toward the local Maxwellian quickly, and this is how [5] defined the strong AP property. i,

22 THE VPFP SYSTEM WITH UNCERTAINTY 53 Fig.. Example 7.: The l -norm of Ef M eq. We take x 0,, Nx = 000, v [ 6, 6], Nv = 400, t [0, 0.0], δ t = δ x/5 and ɛ = 0 3, 0 4, 0 5, K = 4. Left: second order scheme with equilibrium initial data defined as 5. Right: second order scheme with nonequilibrium initial data defined as 6. Fig. 3. ɛx given in Statistical quantities. In this section, we will see the expectation and standard deviation of ρt, x, z, Et, x, z, t, x, z for different cases Mixing regimes. In the first case, we compare the second order gpc-sg method with the reference solution calculated with 0 Legendre quadrature points and mesh size δ x = /000, δ t = δx 5, δ v = 400. The mixing regime is defined as follows: tanh5 0x + tanh5 + 0x, x 0.3, 9 ɛx = 0 3, x > 0.3. Thus it contains both the kinetic and high field regimes. See Figure 3. The initial condition is given by 30 3 ρ 0 = π 6 + sinπx + 0.z, f 0 = ρ 0x, z e v+ xφx,z, periodic in x,, π

23 54 YUHUA ZHU AND SHI JIN with 3 h 0 = ecosπx + 0.z, where the certain part of the initial data is given in [7, section 3.3]. The time evolution of the expectation and the standard deviation for ρ,, E at T = 0., 0., 0.3 are shown in Figure 4. Fig. 4. Example The dotted lines represent the result obtained by gpc-sg: N x = 8, v [ 6, 6], N v = 64, and δ t = δx, K = 5. The solid lines are reference solution with Nx = 000, 5 N v = 400, δ t = δx, and 0 Gaussian quadrature points. 5

24 THE VPFP SYSTEM WITH UNCERTAINTY 55 Figure 4 shows the expectation and deviation of ρ,, and φ at time T = 0., 0., 0.3. One can see the statistic quantities of gpc-sg match well with the reference solution Piecewise constant initial data. In the second case, we test the second order scheme with periodic piecewise constant initial data defined as follows, where the certain part is the same as in [7, section 3.4]. 33 ρ 0, h 0 = 8, + λ z, 0 x < 4 4, ρ 0, h 0 =, + λ z, 8 4 x < 3 4, ρ 0, h 0 = 8, 3 + λ z, 4 x <, f 0 = ρ 0x, z e v+φxx,z, ɛ = 0 3. π Fig. 5. Example 7.3.: The dashed line is the expectation of two cases, while the solid line is obtained by E ± Sd. N x = 64, v [ 6, 6], N v = 00, and δ t = δx 5, K = 5.

25 56 YUHUA ZHU AND SHI JIN Fig. 6. Example 7.3.: The dashed line is the expectation of the two cases, while the solid line is obtained by E ± Sd. N x = 00, v [ 6, 6], N v = 00, and δ t = δx 5, K = 5. In order to test how the random variables affect the final result, we compare two cases:. λ = 0, λ = 0. versus λ = 0, λ = 0... λ = 0, λ = 0. versus λ = 0., λ = 0.. Figure 5 shows the comparison of the first case at T = 0.. As the coefficient of z getting bigger, the expectation remains the same, while the standard deviation becomes bigger and it increases in the same order as the coefficient. Figure 6 shows the comparison of the second case at T = 0.. One can tell that the randomness in the Poisson equation doesn t have a significant effect on density, while it does affect the electric field. Appendix A. The proof of Lemma 3.. Proof.. The conclusion holds for l = m, since from the last line of 4, 34 0 < A m m = C φ m! m b. m!

26 36 THE VPFP SYSTEM WITH UNCERTAINTY 57. Assume the conclusion holds for l = k +,..., m ; then one has m 0 < A m k = C φ 35 i k A m i + m k i=k+ b m b m i i! m! m! + k!i k! i! k!m k! = i=k+ m! k! = b m k m! k! b m k m! k! m i=k+ m k i= b m+ i i k! b i i! b i 4 i Appendix B. The proof of Proposition 6.. m! b m k. k! Proof. To prove a, by the definition of e A = n=0 n! An, one has v M = n= n! va n. One notes v A = P, which implies, v A A = A v A. Therefore, 37 Thus, 38 v A n = n n A n i v AA i = v A A n i A i = n v AA n. i= v M = n= i= n! P An = P M. To prove b, as long as matrices A and B are commutative, then e A e B = e A+B. Since e A e A = e 0 = I, the inverse of M exists and is 39 M = exp A. To prove c, since P is a symmetric matrix, there exists a unity matrix Q and a diagonal matrix Λ = diagλ,..., λ K such that P = Q ΛQ, so P = Q Λ Q. Since P M = e = Q e Λ Q, the eigenvalues of M are e λ m > 0, m =,..., M. The proof for M is similar. To prove d, let P = K xφ k E k + v I K, P = K xφ k E k + v I K ; then it is easy to check P P = P P, hence P P = P P, which means P and P are commutative. Thus 40 Mv Mv = e P P

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