The Vlasov-Poisson-Fokker-Planck System with Uncertainty and a One-dimensional Asymptotic Preserving Method

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1 The Vlasov-Poisson-Fokker-Planck System with Uncertainty and a One-dimensional Asymptotic Preserving Method Yuhua Zhu and Shi Jin August 7, 06 Abstract We develop a stochastic Asymptotic Preserving s-ap scheme for the Vlasov-Poisson- Fokker-Planck VPFP 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 later time, which allows us to establish the spectral convergence of the gpc-sg method. We follow the framework developed in [5] 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. 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 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 R N with velocity v R N. In addition to the classical difficulty of high dimensionality This work was partially supported by NSF grants DMS-584 and DMS-079: RNMS KI-Net, by NSFC grant No , 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 53706, USA yzhu3@wisc.edu Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA sin@wisc.edu, and Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 0040, China

2 to solve equations in the phase space, the problem under study has two more computational challenges: Multi-scale 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, 0]. One successful numerical strategy to efficiently compute into such asymptotic regimes is to develop Asymptotic-Preserving AP schemes, which preserves the continuous 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 [5, 6, 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, 3, ], although its applications in kinetic problems are scarce, see recent efforts in [7,, 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 [7], 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 [5] 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 multi-dimensional s-ap schemes for the VPFP system.

3 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 is t f + v x f ɛ x φ v f = ɛ v [vf + v f],. x φ = ρ h, t > 0, x R N, v R N, z I z, with the following initial condition: f0, x, v, z = f 0 x, v, z, x R N, v R N, 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 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 f 0 x, v, zdxdv = R N R N hx, zdx, R N.3 and the density function ρt, x, z is defined as ρt, x, z = ft, x, v, zdv. R N.4 Besides, we define operators L, L φ as, Lf, φ = t f + v x f ɛ x φ v f ɛ v [vf + v f],.5 L φ f, φ = x φ ρ h..6. The High Field Limit Here we will show the formal limit of. when ɛ 0. First, integrate. over v, t fdv + x vfdv R v x φfdv = v vf + v fdv. N R ɛ N R ɛ N R N.7 Define the flux = vfdv. R N.8 After integrating by parts, one has t ρ + x = 0..9 Then multiply v to both sides of. and integrate over v, ɛ t vf + ɛ vv x f = v v f x φ + v v [vf + v f]. R N R N R N.0 3

4 Let ɛ 0 it becomes which implies, Therefore, one has, 0 = v [ v f x φ + vf + v f] dv, R N. 0 = f x φ + vf + v fdv. R N. = ρ x φ..3 Finally plugging.3 into.9, one gets the high field limit of system., { t ρ x ρ x φ = 0, x φ = ρ h..4 For each fixed z, the rigorous proof for the high field limit of VPFP system in one dimension can be found in [0, 0]. 3 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 5.3. 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, < f, g > π := fgπz dz, 3. I z and weighted norm in x, v, z space f π = I z R N R N f πzdxdvdz 3. Regularity of Solution in the Random Space. 3. 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 where D l = a l C f l!, a = max{c φ, }, G l = l +, l zft π D l e G l t ɛ, for l = 0,, m, 3.3 Proof. For notation simplicity, we take N =. However, the proof can be easily extended to multi-dimensional x and v. First, multiply fπz to. and integrate it over x, v and z, after integration by parts, one gets, ɛ t f π = f π v f π

5 For l =,, m, take l-th derivative in z to., one gets, 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 zf l + v zf. l 3.5 Multiplying πz l zf and integrating over x, v and z, then the second and third terms vanish, so one has, for l =,, m l ɛ t zft l π = l i z l i x φ v zf i zfπz l dxdvdz + zf l π v zf l π. I z R N R N Using Young s Inequality and the boundedness of l z x φ, one gets, 3.6 l ɛ t zft l π Cφ l i v zf i π + l + zft l π v zf l π. 3.7 Multiplying a constant A m l to 3.7 and summing l from to m, then adding A m gives, m ɛ t A m l zft l π m l m m A m l l i v zf i π + l + A m l zft l π A m l v zf l π Cφ l= m = m l=i+ C φ l i A m l A m i v i zf π A m m v i zf π + m l + A m l zft l π. Let A m m = and m l=i+ C φ l i A m l A m i = 0, for i = 0,, m, 3.8 becomes m m ɛ t A m l zft l π l + A m l zft l π. 3.9 and one has a linear system for A m i C φ, i = 0,, m : 0 0 m 0 C φ m..... C φ m m C φ A m 0 A m. A m m A m m Lemma 3.. Solving the linear system 3.0, one has, = m 0 m. m m m m < A m l b m l m! l!, where b = max{, C φ} 3. Proof. See Appendix A. 5

6 Therefore, by Lemma 3., and apply Gronwall s Inequality to 3.9, 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 which implies, where a = max{c φ, } l= 7 3 bm m! e m+t ɛ C f, 3. m z ft π a m m!e m+t ɛ C f, 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, l z v ft π C l e L l ɛ t, for l = 0,, m, 3.4 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 l ɛ t z l v f + ɛv x z l v f + ɛ z l x f l i z l i x φ z i vf x φ v z l v f = v l zf + v l z v f + l z vf ; 3.5 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 i x z l i φ v z i x f = v v z l x f + v z l x f. 3.6 l Multiplying πz l z v f to 3.5 and πz l z x f to 3.6, and integrating over x, v and z, one has respectively, ɛ t z l v f π + R N R N ɛ < l z x f, l z 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 ; 3.7 l ɛ t z l x f π < l i x z l i φ z i v f, z l x f > π dxdv R N R N l R N R N < l i x z l i φ z i v x f, z l x f > π < xφ z l v f, z l x f > π dxdv = l z x f π l z v x f π

7 By Young s Inequality, one gets, l ɛ t z l v f π ɛ z l x f π + ɛ l z l v f π + Cφ l i z i vf π z l vf π ; 3.9 l ɛ t z l x f π C φ + + l z l x f π + C φ z l v f π + Cφ l i z i v f π l + Cφ l i z i v x f π z l v x f π. 3.0 Summing the two inequalities yields, ɛ t l z v f π + z l x f π ɛ + C φ l z l v f π + z l x f l π + C φ l i z i v f π l l + Cφ l i z i vf π z l vf π + Cφ l i z i v x f π z l v x f π. 3. Similarly, for l = 0, one has, 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. 3. Multiplying A m l ɛ t m A m l to 3.0 and Summing it from to m over l, then adding A m 0 3., gives, l z v f π + l z x f π m ɛ + C φ l A m l + m Cφ l=i+ Let A m m = and A m i ɛ t m l i A m l A m i m A m l l z v f π + l z x f π m + Cφ l=i+ m l i A m l z i v f π i z vf π + i z v x f π. 3.3 solves 3.0, for i = 0,, m, one has, l z v f π + l z x f π m ɛ + C φ l A m l m ɛ + C φ l A m l l z v f π + l z x f π m + A m i i z v f π l z v f π + z l x f π, 3.4 then by Lemma 3. and Gronwall s Inequality, one obtains, m A m l l z v f π + z l x f 7 π 3 bm m! e ɛ+cφ+5+m ɛ t Cf. 3.5 Therefore, one can get, which completes the proof. m z v f π 3a m m!e ɛ+c φ +5+m ɛ t C f

8 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 and v f remain in H m at later time. Thus the regularity in z of the initial data is preserved in time. 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, W K π = {g : I z R : g span{φ k z} K }, 4. where Φ k, k = 0,, K is a set of d-variate orthonormal polynomials of degree k satisfying, < Φ k, Φ l > π = EΦ k Φ l = Φ k zφ l zπzdz = δ kl. 4. I z Here E means the expected value, and δ kl is the Kronecker delta function. By the classical approximation theory, Wπ is a Hilbert space with inner product <, > π. Thus the solution ft, x, v, z, φt, x, z to. can be represented as ft, x, v, z = f k t, x, vφ k z, φt, x, z = φ k t, xφ k z, in L π. 4.3 In the gpc stochastic Galerkin 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, K ˆf K t, x, v, z = ˆf k t, x, vφ k z ˆf K K Φ K, ˆφK t, x, z = ˆφ k t, xφ k z ˆφ K Φ K, where Φ K = Φ 0,, Φ K, and ˆf k =< ˆf K, Φ k > π, ˆφ k =< ˆφ K, Φ k > π, which are independent of z, are the components of vector ˆf K, ˆφ K satisfying, for 0 K, We also approximate the given charge h by < L ˆf K, ˆφ K, Φ > π = 0, 4.4 < L φ ˆf K, ˆφ K, Φ > π = ĥ K x, z = where ĥkx =< h, Φ k > π, for k = 0,, K. By the definition of ρ in.4, the numerical approximation of ρ is, ˆρ K t, x, z = K ĥ k Φ k ĥk Φ K, 4.6 K ˆρ k Φ k ˆρ K Φ K, 4.7 8

9 where ˆρ k t, x = ˆf R N k t, x, vdv, for k = 0,, K. By equation 4.5, we have for each = 0,, K, 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, 4.8 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, also for the sake of combining the stiff terms and forming an AP scheme as in [5], we give the following Lemma. Lemma 4.. For matrix E i, 0 i K, defined above, one has [ K K ] x φ k v f l E kl = v E k ˆf K x φ k. 4.9 k, Proof. 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 E k l vi f l = [ K N K ] xi φ k vi E k l f l k, l= K N K = xi φ k vi E k ˆf K = v [ x φ E k ˆf K,..., xn φ k E k ˆf K ] [ K ] = v E k ˆf K x φ k. 4.0 k= Now by Lemma 4., 4.8 can be written in a vector form as [ t ˆf K + x ˆf K v K ] ɛ v E k ˆf K x φ k = [ ɛ v ˆf K v K + v ˆf ], ˆφK x = ˆρ K ĥk 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 t > 0, ˆf K t π e 3Nt ɛ ˆf K 0 π 5. 9

10 Proof. Due to the orthogonality of φ k z, one has ˆf K π = ˆf K L, with L defined as, L = R 3 R N dxdv, 5. where is the regular Euclidean norm for vectors. Therefore one only needs to prove the theorem for ˆf K t L. Multiplying ˆf to 4.8 and integrating over x and v, R N = ɛ R N R N t ˆf R N v v + v x ˆf ɛ ˆf K k,l, xi φ k ˆf vi ˆfl E kl dxdv dxdv + N ɛ ˆf L ɛ ˆf v L, 5.3 After integration by parts the second term on the LHS vanishes, and the first term of the RHS becomes N ɛ R ˆf N R N dxdv. Sum from to K, one gets, t ˆf K L ɛ R N R N K k,l,i,=0 xi φ k ˆf vi ˆfl E kl dxdv Note the second term on the LHS also vanishes, since = ɛ = ɛ ɛ R N R N R N R N R N K k,l,i,=0 K k, =0 K R N k, =0 xi φ k ˆf vi ˆfl E kl dxdv K xi φ k E k vi ˆf K xi φ k E k vi ˆf + + K k, l K k, >l N ɛ + N ɛ ˆf K L. 5.4 K xi φ k ˆf vi ˆfl E k l dxdv, K xi φ k E k l vi ˆf ˆfl dxdv. 5.5 By the symmetric of E k, where the last inequality uses the symmetry of E k. Both terms in 5.5 vanish after integration by parts, so 5.4 implies, t ˆf N K L ɛ + N ɛ ˆf K L, 5.6 By Gronwall s Inequality, which completes the proof. ˆf K t L e 3Nt ɛ ˆf K 0 L, 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 multi-dimensional z. We define operators L f, K as, L f := ɛ t + ɛv x v v N v, K := x φ v, then L = L f K

11 Let the proection of the exact solution ft, x, v, z to the subspace W K π be P K f, K K P K f := < f, Φ k > π Φ k z := f k t, x, vφ k z := f K Φ K, 5.9 where f = f 0,, f K. As defined in 4.4, 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 + P K f ˆf K := R K + µ K, 5.0 Where R K = f k t, x, vφ k z, 5. k=k+ is the proection error. Define vector µ K = µ 0,, µ K with µ i = f i ˆf i, i = 0,, K. 5. 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 z l v f0 π C f, z l x φ L C φ, z l xφ L C φ, for l = 0,, m, then for 0 < t < T, µ K t π H me Lm+3N ɛ t K m, 5.3 where H m = C AC mc φ Lm, with C A a constant depending on polynomials {Φ k z 0 k m}. Proof. Subtracting < Lf, Φ K > π = 0 by < L ˆf K, Φ K > π = 0, one has Since L f is independent of z, Plugging 5.5 into 5.4 gives, < L f f ˆf K, Φ K > π < Kf ˆf K, Φ K > π = < L f f ˆf K, Φ K > π =L f < f ˆf K, Φ K > π = L f µ K. 5.5 L f µ K < Kµ K + R K, Φ K > π = Taking dot product of µ K to 5.6, then integrating over x, v, yields, [ 0 = Lf µ K µ K < Kµ K + R K, Φ K > π µ K] dxdv R N R N =ɛ t µ K π N µ K π + v µ K π < KR K, µ K > π dxdv R N R N x φ v µ K πz dzdxdv R N R N I z =ɛ t µ K π N µ K π + v µ K π < KR K, µ K > π dxdv. 5.7 R N R N

12 This gives, ɛ t µ K π N + N µ K π + C φ v R K π. 5.8 Since µ K 0 π = u K 0 dxdv = 0, and by Grownwall s inequality implies, µ K t π ɛ C φ t By classical approximation theory and Theorem 3.3, v R K π C A m z v f π K m 0 v R K s πds e 3N ɛ t. 5.9 C Lm AC m e ɛ t K m, 5.0 where C A is a constant depending on polynomials {Φ k z 0 k m}. Plugging 5.0 into 5.9 yields, where H m = C AC mc φ Lm, which implies, µ K t π H me Lm t ɛ e 3N ɛ t, 5. K m µ K t π H me Lm+3N ɛ t K m. 5. Theorem 5.3. Given φt, x, z, if for some integer m > 0, and positive constants C f, C φ, such that l zf0 π C f, l z v f0 C f, l z x φ L C φ, l z xφ L C φ, for l = 0,, m. Then the K-th order numerical approximation ˆf K converges to the solution f with an error, f ˆf K π O m K m. 5.3 where O m = C A D m e Gm ɛ ɛ. t + H m e Lm+3N ɛ t is a finite positive constant depending on C f, C φ and Proof. f ˆf K π R K π + µ K π C A m z f π K m C AD m e Gm ɛ + H me Lm+3N ɛ t K m t + H m e Lm+3N ɛ t K m, 5.4 The first inequality is because of the definition in 5.0, 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 as ɛ 0, one needs K Oe mt ɛ to get a good accuracy. This motivates the development of the s-ap scheme in which one can take K independent of ɛ.

13 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 4.. Integrating 4., and letting ĵk = R ˆf K v dv be the flux, one gets, t ˆρ K + x ĵk = 0, 6. then, multiplying v, the transpose of v, to 6.4 and integrating it over v gives, K E k ˆρ K x φ k + ĵk = Plugging 6. into 6. 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, 6.3 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. 6. The fully discrete first order scheme Here we ll give the VPFP system with [ uncertainty a fully discrete scheme when N =. K ] K First we combine the stiff terms v x ˆφk E k ˆf and v v ˆf K K + v ˆf, then [ t ˆf K + v x ˆf K = K ] ɛ v x ˆφk E k + vi K ˆf K K + v ˆf, xx φ K = ˆρ K ĥk. where I K is K K identity matrix. Here we denote, F = 6.4 K x ˆφk E k, P = F + vi K, A = P, 6.5 where Let P := P P. 6.6 M = π e A. 6.7 Concerning the properties of the matrix M, we give the following proposition. Proposition 6.. Suppose M is defined in 6.7, then 3

14 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 the Appendix A. Back to system 6.4, where the stiff terms can be represented by v [ M v M ˆf K ] from Proposition 6. a, b, g, thus 6.4 is equivalent to t ˆf K + v x ˆf 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 mesh points in x and v directions respectively. Let x i = a+iδ x, v = δ v, ˆρ n i 6.8, n 0. N x, N v even are numbers of = δ Nv/ ˆf v n = N v/ i, be the numerical approximation of density ˆρ. We choose N v sufficiently large such that outside the velocity domain, f v Nv δ v 0, M v Nv δ v 0, 6.9 during the computational time. We basically adopt the scheme in [5] for deterministic problem. The first order scheme is ˆf n+ i ˆf n i δ t + ˆf n i+, ˆf n i, = n+ P ˆf i, 6.0 δ x ɛ ˆφn+ x i = ˆρ n+ i ĥn+ i, 6. where the upwind flux is used for spatial discretization, P ˆf n i+, = v + v ˆf n i, + v v ˆf n i+,. 6. n+ ˆf i is the discretization form of P ˆf = v [M v M ˆf], which is defined as, P ˆf = δ v [ M +/ [ v M ˆf]+/ M / [ v M ˆf] / ] = δ v = M / δv [ M / + M / M ˆf + + M ˆf M / [ M / + ˆf + M / + + M / The algorithm is implemented as following: M / M / M M / ˆf M ˆf ] ˆf + M / ˆf ]

15 Step. Summing 6.0 over. Since the RHS vanishes, one gets, 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, 6.4 Step. By using a Poisson solver, one gets as, M n+ i, = K exp π n+ ˆφ i from 6., which in term gives M n+ i ˆφ k n+ i+ ˆφ k n+ i E k + v I K. 6.5 δ x Step 3. Since Fi n = K ˆφ k n i+ ˆφ k n i δ x E 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 Let Λ n i = e 4 v+λn i, 6.3 can be written as, P { n+ QΛ [ ˆf i = ɛ v Λ + Q ˆf+ Λ + + Λ Λ Λ Multiply Λ n+ i Q n+ i to 6.0, and let ĝ n+ i = Λ n+ i Q n+ i Let b n i = Λn+ i Q n+ i [ ĝ n+ i,+ Λ n+ i,+ + Λ i, Λ i =ɛδvλ n+ i Q n+ i ˆf n i+, ˆf n i, component ĝ n+ k of ĝ n+, k = 0,, K, [ mk g k n+ i,+ n+ i,+ m k n+ i δ x + ɛ v t ˆf n i+, ˆf n i, ˆf n i δ t + m k n+ i, m k n+ + e v t i n+ δ x ] ] } n+ Q ˆf + Λ Q ˆf ] ĝ n+ i + ĝ n+ i, ˆf n i δ t n+ ˆf i, one has, i , then one has a scalar solver for each g k n+ i + g k n+ i, = b k n i, 6.8 where m k n+ i = e v +λ k i 4, which has been proved in [8] that the linear system for g k n+ i is positive definite, so one can invert it by 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,, thus for fixed i, n, 6.3 will become, P ˆf = [ M+ + M δv M + ˆf + M ˆf M ] + M M ˆf M ˆf = Q [ Λ δv + + Λ ĝ + Λ + + Λ + Λ ĝ + Λ + Λ ] ĝ 6.9 5

16 Thus, 6.7 becomes, [ Λ + + Λ ĝ + Λ δ vλ + Λ ĝ + Λ + Λ ĝ ] n+ =δ vq ˆf n i δ t + ˆf n i+, ˆf n i, 6.0 δ x which can be decomposed to a scalar solver for each component of ĝ n+. Besides, it is easy to see the coefficient in 6.0 is diagonally dominated matrix with negative diagonal entries, so it is a negative definite matrix. i 6.3 The s-ap property 6.3. Mass Conservation Since P ˆf has the property of mass conservation, its discretization P ˆf should have the same property. Let K = M / + M / M ˆf + + M ˆf, 6. then, by 6.3, P ˆf = δ [K K ] = v δv K δ K v = δv K δv K = Thus, summing 6.0, one can get the scheme for ˆρ n+, 6.4, which also implies i ˆρn+ i = i ˆρn i The formal proof of s-ap Here we want to prove the scheme is stochastic asymptotic preserving, 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 6.0, ˆf n i M n iĉn i, as ɛ 0, where ĉn i is independent of. Proof. For fixed i, n, let ɛ 0, multiply v to 6.0 and sum it over, one gets, 0 = v P ˆf = δv v [K K ] = δv δ v K, 6.3 which is equivalent to, K = Letting ɛ 0, 6.0 also implies P ˆf = 0 for, or equivalently, K K = 0 for. 6.5 δ v 6

17 This implies, K = c for, 6.6 where c is a constant depending on i and n. From 6.4, 6.6, one has K 0. By the definition of K in 6., this implies, therefore, this gives, M n i,+ ˆf n i,+ M n i ˆf n i = 0, 6.7 M n i ˆf n i = ĉ n i for, 6.8 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. Proof. As defined in section 6., δ v = Nv = Nv = Nv ˆf i = ˆρ i = δ v = Nv M i ĉ i. 6.9 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, M = Q e Λ +v Λ+v I Q = Q diag π π Use the trapezoidal rule and assumption 6.9, = e λ i +v dv π = R = Nv = Nv + e λ i +v + exp π π Again by assumption 6.9, so 6.3 implies, π exp e λ +v λ i + v Nv λ i + v Nv +,, e λn+v + exp π π exp Q λ i + v Nv + Oδ v, 6.3 λ i + v Nv Oδ v, 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 =Q + Oδ viq = + Oδ v I

18 Therefore, δ v M = + Oδv I = + Oδv I So by 6.9 and 6.34, 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. Proof. From Lemma 6.4 and 6.3, as ɛ 0, Thus, F + = = v + v Mvdv = R F ˆf i n Mi n ˆρ n i + Oδv e A da F π F 0 vmvdv = 0 π vi K + F e A dv 0 F Mvdv P e dp = F e F erff, 6.36 π π where erfx = x π e t dt, F, P, A is defined in 6.5. Similarly, F v v 0 = Mvdv = vmvdv = F e F erf F R π Then F n i+ defined in 6.4 becomes F n i+ = F + ˆρ n i + F ˆρ n i+ + Oδ v, 6.38 which is exactly the numerical flux of the kinetic scheme for 6.3 by [7], ch3. So as ɛ 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 s-ap property. 6.4 A second order scheme Using backward difference formula for time discretization [9], and MUSCL scheme for space discretization, the second order scheme is given by 3 ˆf n+ i 4 ˆf i n δ t + ˆf n i n n + v x ˆf i v x ˆf i = n+ P ˆf i, 6.39 ɛ ˆφn+ x i = ˆρ n+ i ĥn+ i by Poisson Solver Here, v x ˆfi = v ˆfi+, ˆf i, δ x, and ˆf i+, = ˆf i, + ψθ+ ˆf i+ i+ ˆf i v > 0, ˆf i+, = ˆf i+, ψθ+ ˆf i+ i+ ˆf i v <

19 Where θ + = ˆf i ˆf i i+ ˆf i ˆf and θ = ˆf i+ ˆf i+ i i+ ˆf i+ ˆf are smooth indicators, ψ = max0, min, θ is the i slope limiter function [9]. The AP property can be similarly established as the first order scheme, so we omit the details here. 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], + λ z xx φ = ρ h, x [x 0, x I ], v R, 7. with periodic function φt, x, z satisfying, φt, x 0, z = φt, x I, z = and only in Section 7.3., λ 0. Initial conditions are given by, ρ 0 = ρ 0 x, λ z, f 0 = f 0 x, v, λ z, 7.3 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, standard deviation are retrieved as, E[ ˆf K ] = ˆf 0, S[ ˆf K ] = K ˆf m 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: ρ 0 x, z = + sinxe z, f 0 = ρ 0x, z e v+ xφx,z, x 0, π. 7.5 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 7.6 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, 7.7 i, where f, the reference solution, is calculated by the Stochastic Collocation method [] 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 K-th order gpc-sg and the same mesh size as the reference solution. m= i, 9

20 Figure : Example 7.: Error of the numerical solution at T = 0.0 defined in 7.7 when ɛ =, 0 3, 0 5. We take δ x = π δx, v [ 6, 6], δv =, δt =, 0 K 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. 7. The asymptotic preserving property This section is devoted to check the asymptotic preserving property of the scheme. We take the equilibrium initial data, and non-equilibrium initial data respectively. The certain part of the initial data in this example is same as section 3. in [5]. π ρ 0 x, v, z = + cosπx + λ z, hx, z = ecosπx + 0.z, x [0, ]. 7.8 For equilibrium initial condition, f 0 is given by, f 0 x, v, z = ρ 0x, z e v+ xφ, periodic in x [0, ], 7.9 π while for the non-equilibrium initial data, f 0 is given by, e v+.5 + e v.5 f 0 x, v, z = ρ 0x, z π We study the evolution of the difference between f and equilibrium M eq =, periodic in x [0, ]. 7.0 ρ e v+ xφ, with π respect to different ɛ as shown in Figure. Here the difference is defined as, difference = Ef EM eq = δ x δ v Ef i EM eq i 7. Figure shows the time evolution of the difference defined in 7. with different ɛ. One can see no matter whether the initial data is equilibrium or non-equilibrium, the s-ap method will push f towards the local Maxwellian quickly, and this is how [5] defined strong AP property. i, 0

21 Figure : 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. The left figure: second order scheme with equilibrium initial data defined as 7.9; The right figure: second order scheme with non-equilibrium initial data defined as 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 following, tanh5 0x + tanh5 + 0x, x 0.3, ɛx = 0 3, x > So it contains both the kinetic and high field regimes. See Figure 3 The initial condition is given by, π ρ 0 = 6 + sinπx + 0.z, f 0 = ρ 0x, z e v+ xφx,z, periodic in x,. 7.3 π with h 0 = ecosπx + 0.z. 7.4 Where the certain part of the initial data is given in [5] Section 3.3. The time evolution of the expectation and standard deviation for ρ,, E at T = 0., 0., 0.3 are shown in Figure 4. 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 matches well with the reference solution.

22 Figure 3: ɛx given in Piecewise Constant Initial Data In the second case, we test the second order scheme with periodic piecewise constant initial data defined as following, where the certain part is same as [5] Section 3.4. ρ 0, h 0 = 8, 4 + λ z, 0 x < 4, ρ 0, h 0 =, 8 + λ z, 4 x < 3 4, ρ 0, h 0 = 8, + λ 3 z, 4 x <, f 0 = ρ 0x, z e v+φxx,z, ɛ = π In order to test how the random variables affect the final result, we compare two cases,. λ = 0, λ = 0.; v.s. λ = 0, λ = 0... λ = 0, λ = 0.; v.s. 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. A Appendices A. The proof of Lemma 3. Proof.. The conclusion holds for l = m, since from the last line of 3.0, 0 < A m m = C φ m! m b. A. m!

23 Figure 4: Example The dot lines represent for 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 3

24 Figure 5: Example 7.3.: The dash 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. 4

25 Figure 6: Example 7.3.: The dash 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. 5

26 . Assume the conclusion holds for l = k +,, m, then one has, m 0 < A m k = C φ i k A m i + m k b m b m i i! k!i k! = m! k! i=k+ b m k m! k! i=k+ m! + i! m! k!m k! m m k m! b m+ i = b m k b i i k! k! i! i=k+ i= m! b i 4 i b m k. A. k! A. The proof of Proposition 6. Proof. To prove a, By the definition of e A = n=0 n! An, one has, v M = n= One notes v A = P, which implies, v A A = A v A. Therefore, n! va n. A.3 n n v A n = A n i v AA i = v A A n i A i = n v AA n. i= i= A.4 Thus, v M = n= n! P An = P M. A.5 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 M = exp A. A.6 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 Mv Mv = e P P. A.7 is symmetric. Since if the matrices A, B are positive definite and AB is symmetric, then AB is still positive definite. Therefore, we conclude Mv Mv is still positive definite. The commutativity can be easily obtained from A.7. 6

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