Approximation and Fast Cacuation of Non-oca Boundary Conditions for the Time-dependent Schrödinger Equation Anton Arnod, Matthias Ehrhardt 2, and Ivan Sofronov 3 Universität Münster, Institut für Numerische Mathematik, Einsteinstr. 62, D-4849 Münster, Germany (http://www.math.uni-muenster.de/u/arnod/) 2 Technische Universität Berin, Institut für Mathematik, Str. des 7. Juni 36, D-623 Berin, Germany (http://www.math.tu-berin.de/~ehrhardt/) 3 Kedysh Institute of Appied Mathematics, Russian Academy of Sciences, Moscow, Russia (sofronov@spp.kedysh.ru) Summary. We present a way to efficienty treat the we-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: firsty, to derive a discrete transparent boundary condition (DTBC) based on the Crank-Nicoson finite difference scheme for the governing equation. And, secondy, to approximate the discrete convoution kerne of DTBC by sum-of-exponentias for a rapid recursive cacuation of the convoution. We iustrate the efficiency of the proposed method on severa exampes. A much more detaied version of this artice can be found in Arnod et a. [23]. Introduction Discrete transparent boundary conditions for the discrete D Schrödinger equation ir(ψ j,n+ ψ j,n ) = 2 (ψ j,n+ + ψ j,n ) wv j,n+ 2 (ψ j,n+ + ψ j,n ), () where 2 ψ j = ψ j+ 2ψ j + ψ j, R = 4 x 2 / t, w = 2 x 2, V j,n+ := 2 V (x j, t n+ ), x j = j x, j Z; and V (x, t) = V 2 = const. for x ; V (x, t) = V + = const. for x X, t, ψ(x, ) = ψ I (x), with supp ψ I [, X], were introduced in Arnod [998]. The DTBC at e.g. the eft boundary point j = reads, cf. Thm. 3.8 in Ehrhardt and Arnod [2]: ψ,n s ψ,n = n k= s n kψ,k ψ,n, n. (2) The convoution kerne {s n } can be obtained by expicity cacuating the inverse Z transform of the function ŝ(z) := z+ ˆ z (z), where ˆ (z) = iζ ± ζ(ζ + 2i), ζ = R z 2 z+ + i x2 V (choose sign such that ˆ (z) > ).
42 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov Using (2) in numerica simuations permits to avoid any boundary refections and it renders the fuy discrete scheme unconditionay stabe, ike the Crank-Nicoson scheme () for the whoe-space probem. However, the numerica effort to evauate the DTBC increases ineary in t and it can sharpy raise the tota computationa costs. A strategy to overcome this drawback is the key issue of this paper. 2 Approximation by Sums of Exponentias The convoution coefficients s n appearing in the DTBC (2) can either be obtained from (engthy) expicit formuas or evauated numericay: s n ρ n N N k= ŝ(ρeiϕ k )e inϕ k, n =,,...,N. Here ϕ k = 2πk/N, and ρ > is a reguarization parameter. For the question of choosing ρ we refer the reader to Arnod et a. [23] and the references therein. Our fast method to cacuate the discrete convoution in (2) is based on approximating these coefficients s n by the foowing ansatz (sum of exponentias): s n s n := { sn, n =,...,ν, L = b q n, n = ν, ν +,..., where L, ν IN are fixed numbers. In order to find the appropriate constants {b, q }, we fix L and ν in (3) (e.g. ν = 2), and consider the Padé approximation P L (x) Q L(x) for the forma power series: f(x) := s ν + s ν+ x + s ν+2 x 2 +..., x. Theorem. Let the poynomia Q L (x) have L simpe roots q with q >, =,...,L. Then s n = L = (3) b q n, n = ν, ν +,..., (4) where b := P L (q ) Q L (q ) qν, =,...,L. (5) Remark. A our practica cacuations confirm that the assumption of Theorem hods for any desired L, athough we cannot prove this. Remark 2. According to the definition of the Padé agorithm the first 2L+ν coefficients are reproduced exacty: s n = s n for n = ν, ν +,...,2L + ν. For the remaining s n with n > 2L + ν, the foowing estimate hods: s n s n = O(n 3 2). A typica graph of s n and s n s n versus n for L = 2 is shown in Fig. (note the different scaing for both graphs).
Efficient Non-oca Boundary Conditions for the Schrödinger Equation 43 s n and error s n s ~ n 3 4 2 s n 5 error s n s ~ n 4 6 6 2 3 4 5 6 7 8 7 n Fig.. Convoution coefficients s n (eft axis, dashed ine) and error s n s n of the convoution coefficients (right axis); x = /6, t = 2 5, V (L = 2). 3 The Transformation Rue A nice property of the considered approach consists of the foowing: once the approximate convoution coefficients { s n } are cacuated for particuar discretization parameters { x, t, V }, it is easy to transform them into appropriate coefficients for any other discretization. We sha confine this discussion to the case ν = 2: Transformation rue 3. For ν = 2, et the rationa function ˆ s(z) = s + s z + L = b q z q z (6) be the Z transform of the convoution kerne { s n } n= from (3), where { s n } is assumed to be an approximation to a DTBC for the equation () with a given set { x, t, V }. Then, for another set { x, t, V }, one can take the approximation where ˆ s (z) := s + s z + L b q = z q z, (7) q := q ā b a q b, b := b aā b b + q q (a q b)(q ā b), (8) + q
44 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov a := 2 x2 t + 2 x2 t + i( x 2 V x 2 V ), (9) b := 2 x2 t 2 x2 t i( x 2 V x 2 V ). () s, s are the exact convoution coefficients for the parameters { x, t, V }. Whie the Padé agorithm provides a method to cacuate approximate convoution coefficients s n for fixed parameters { x, t, V }, the Transformation rue yieds the natura ink between different parameter sets { x, t, V } (and L fixed). Exampe. For L = we cacuated the coefficients {b, q } with the parameters x =, t =, V = and then used the Transformation 3. to cacuate the coefficients {b, q } for the parameters x = /6, t = 2 5, V = 45. Fig. 2 shows that the resuting convoution coefficients s n are in this exampe even better approximations to the exact coefficients s n than the coefficients s n, which are obtained directy from the Padé agorithm discussed in Theorem. Hence, the numerica soution of the corresponding Schrödinger equation is aso more accurate (cf. Fig. 5). The Mape code that was used to cacuate the coefficients q, b in the approximation (3) incuding the expicit formuas in Transformation rue 3. can be downoaded from the authors homepages. ~ 7 x error s n s 3 n, s n s~ * n 6 5 4 3 2 5 5 2 25 3 n Fig. 2. Approximation error of the approximate convoution coefficients for ν = 2, x = /6, t = 2 5, V = 45: The error of s n (- - -) obtained from the transformation rue and the error of s n ( ) obtained from a direct Padé approximation of the exact coefficients s n.
Efficient Non-oca Boundary Conditions for the Schrödinger Equation 45 4 Fast Evauation of the Discrete Convoution Given the approximation (3) of the discrete convoution kerne appearing in the DTBC (2), the convoution n ν C (n) (u) := u k s n k, s n = k= L = b q n, q >, () of a discrete function u k, k =, 2,..., can be cacuated efficienty by recurrence formuas, cf. Sofronov [998]: Theorem 2. The function C (n) (u) from () for n ν + is represented by where C (n) (u) = q C (n ) C (n) (u) = L = C (n) (u), (2) (u) + b q ν u n ν for n ν +, C (ν) (u). (3) This recursion drasticay reduces the computationa effort of evauating DT- BCs for ong time computations (n ): O(L n) instead of O(n 2 ) arithmetic operations. 5 Numerica Exampes In this section we sha present two exampes to compare the numerica resuts from using our approach of the approximated DTBC, i.e. the sum-of-exponentias-ansatz (3) (with ν = 2) to the soution using the exact DTBC (2). Exampe 2. As an exampe, we consider () on x with V = V + =, and initia data ψ I (x) = exp(i5x 3(x.5) 2 ). The time evoution of the approximate soution ψ a (x, t) using the approximated DTBC with convoution coefficients { s n } and L =, L = 2, respectivey, is shown in Fig. 3 (observe the viewing ange). Whie one can observe some refected wave when using the approximated DTBC with L =, there are amost no refections visibe when using the approximated DTBC with L = 2. The goa is to investigate the ong time stabiity behaviour of the approximated DTBC with the sum-of-exponentias ansatz. The reference soution ψ ref with x = /4, t = 2 5 is obtained by using exact DTBCs (2) at the ends x = and x =. We vary the parameter L =, 2, 3, 4 in (3), find the corresponding approximate DTBCs, and show the reative error of the approximate soution, i.e. ψa ψ ref L2 (t) ψ I L2. The resut up to n = 5
46 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov..2.8 ψ(x,t).3.6.4.4.5.2.6.9.8.7.7.6.5.4.3.2. t.8 x..2.8 ψ(x,t).3.6.4.4.5.2.6.9.8.7.7.6.5.4.3.2. t.8 x Fig. 3. Time evoution of ψa (x, t) : The approximate convoution coefficients consisting of L = discrete exponentias give rise to a refected wave (upper figure). Using L = 2 discrete exponentias make refections amost invisibe (ower figure). is shown in Fig. 4. Larger vaues of L ceary yied more accurate coefficients and hence a more accurate soution ψa. Fig. 4 aso shows the discretization kψ ψ k (t) error, i.e. refkψi kanl L2, where ψan, the anaytic soution of this exampe is 2 expicity computabe. Exampe 3. The second exampe considers () on [, 2] with zero potentia in the interior (V (x) for < x < 2) and V (x) 45 outside the computa-
Efficient Non-oca Boundary Conditions for the Schrödinger Equation 47 reative L 2 Norm of refected part for different number of coefficients 2 ψ ref L = L = 2 L = 3 L = 4 3 ψ a ψ ref L 2 / ψ I L 2 4 5 6 7..2.3 t Fig. 4. Error of the approximate soution ψ a(t) with approximate convoution coefficients consisting of L =, 2, 3, 4 discrete exponentias and discretization error. ψ ref is the reative error of ψ ref. The error-peak between t =. and t =.2 corresponds to the first refected wave. tiona domain. The initia data is taken as [ψ I (x) = exp(ix 3(x ) 2 )], and this wave packet is partiay refected at the boundaries. We use the rather coarse space discretization x = /6, the time step t = 2 5, and the exact DTBC (2). The vaue of the potentia is chosen such that at time t =.8, i.e. after 4 time steps 75% of the mass ( ψ(., t) 2 2 ) has eft the domain. Fig. 5 shows the time decay of the discrete 2 -norm ψ(., t) 2 and the tempora evoution of the error e L (., t) 2 := ψ a (., t) ψ ref (., t) 2 / ψ I L2 when using an approximated DTBC with L = 2, 3, 4. Additionay, we cacuated for L = 2 the coefficients {b, q } for the normaized parameters x =, t =, V = and then used the Transformation rue 3. to cacuate the coefficients {b, q } for the desired parameters. 6 Concusion For numerica simuations of the Schrödinger equation one has to introduce artificia (preferabe transparent) boundary conditions in order to confine the cacuation to a finite region. Such TBCs are non-oca in time (of convoution form). Hence, the numerica costs (just) for evauating these BCs grow quadraticay in time. And for ong-time cacuations it can easiy outweigh the costs for soving the PDE inside the computationa domain. Here, we presented an efficient method to overcome this probem. We construct approximate DTBCs that are of a sum-of-exponentia form and hence
48 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov.5 ψ(.,t) 2, e L (.,t) 2.4 DTBC L = 2 L = 2 (trafo) L = 3 L = 4.3.2..2.4.6.8..2.4.6 t Fig. 5. Time evoution within the potentia we (V = 45) of ψ(., t) 2 and of the errors e L(., t) 2 that are due to approximated DTBCs with L = 2, 3, 4. L = 2 (trafo) uses coefficients cacuated by the Transformation rue 3.. ony invove a ineary growing numerica effort. Moreover, these BCs yied very accurate soutions, and it was shown in Arnod et a. [23] that the resuting initia-boundary vaue scheme is conditionay 2 -stabe on [, T] as t (e.g. for < t < t and x = x = const.). Acknowedgement. The first two authors were partiay supported by the IHP Network HPRN-CT-22-282 from the EU and the DFG under Grant-No. AR 277/3-. The second author was supported by the DFG Research Center Mathematics for key technoogies (FZT 86) in Berin. The third author was partiay supported by RFBR-Grant No. --52 and by Saarand University. References A. Arnod. Numericay absorbing boundary conditions for quantum evoution equations. VLSI Design, 6:33 39, 998. A. Arnod, M. Ehrhardt, and I. Sofronov. Discrete transparent boundary conditions for the Schrödinger equation: Fast cacuation, approximation, and stabiity. Comm. Math. Sci., :5 556, 23. M. Ehrhardt and A. Arnod. Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma, 6:57 8, 2. I. Sofronov. Artificia boundary conditions of absoute transparency for twoand threedimensiona externa time dependent scattering probems. Euro. J. App. Math., 9:56 588, 998.