Energy behaviour of the Boris method for charged-particle dynamics

Transcription

1 Version of 25 April 218 Energy behaviour of the Boris metho for charge-particle ynamics Ernst Hairer 1, Christian Lubich 2 Abstract The Boris algorithm is a wiely use numerical integrator for the motion of particles in a magnetic fiel. This article proves near-conservation of energy over very long times in the special cases where the magnetic fiel is constant or the electric potential is quaratic. When none of these assumptions is satisfie, it is illustrate by numerical examples that the numerical energy can have a linear rift or its error can behave like a ranom walk. If the system has a rotational symmetry an the magnetic fiel is constant, then also the momentum is approximately preserve over very long times, but in a spatially varying magnetic fiel this is generally not satisfie. Keywors. Boris algorithm, charge particle, magnetic fiel, energy conservation, backwar error analysis, moifie ifferential equation. Mathematics Subject Classification (21): 65L6, 65P1, 78A35, 78M25 1 Dept. e Mathématiques, Univ. e Genève, CH-1211 Genève 24, Switzerlan, Ernst.Hairer@unige.ch 2 Mathematisches Institut, Univ. Tübingen, D-7276 Tübingen, Germany, Lubich@na.uni-tuebingen.e

2 2 E. Hairer, Ch. Lubich 1 Introuction For a particle with position x(t) R 3 moving in an electro-magnetic fiel, Newton s secon law together with Lorentz s force equation gives the secon orer ifferential equation (assuming suitable units) ẍ = ẋ B(x) + F (x). (1.1) Here, F (x) = U(x) is an electric fiel with the scalar potential U(x), an B(x) = A(x) is a magnetic fiel with the vector potential A(x) R 3. We assume throughout this paper that the forces are smooth functions of x. Denoting by v = ẋ the velocity of the particle, the energy is given by E(x, v) = 1 2 v 2 + U(x) (1.2) an it is conserve along solutions of (1.1). The simplest iscretization of (1.1) is the Boris metho [1] x n+1 2x n + x n 1 h 2 = x n+1 x n 1 2h B(x n ) U(x n ). (1.3) It is a symmetric, secon-orer numerical integrator. Its popularity is mostly ue to the fact that it is essentially explicit an has shown a goo long-time behaviour in many examples. It just requires the solution of a 3-imensional linear system which can be efficiently solve by a Roriguez formula. In the absence of the magnetic fiel, it reuces to the Störmer Verlet scheme (see [5, Section I.1.4] or [4]). The two-step formulation (1.3) is very sensitive to roun-off errors. For a practical implementation one introuces a velocity approximation for the ifference x n+1 x n, so that the Boris metho becomes x n+1 = x n + h v n+1/2 v n+1/2 = v n 1/2 + h v n B(x n ) h U(x n ), (1.4) where v n = 1 2 (v n+1/2 + v n 1/2 ) = 1 2h (x n+1 x n 1 ) is taken as the velocity approximation at the gri points. Given the initial values (x, v ), the metho is starte with v 1/2 = v + h 2 v B(x ) h 2 U(x ). It is known from [9] that the map (x n, v n 1/2 ) (x n+1, v n+1/2 ) is volumepreserving. Moreover, if the magnetic fiel B is constant, then with the momenta p n = v n + A(x n ), the map (x n, p n ) (x n+1, p n+1 ) is symplectic. This follows from the fact that the Boris metho is a variational integrator if an only if B is constant [2]. In Section 2 we escribe the backwar error analysis (moifie ifferential equation) for the Boris metho. We use this to show long-time nearconservation of the energy for two particular cases: if the magnetic fiel B is constant (Section 3), if the electric potential U is quaratic (Section 4).

3 Energy behaviour of the Boris metho for charge-particle ynamics 3 In Section 5 we then illustrate by numerical experiments that in the general situation the numerical energy can have a linear rift or its error can behave like a ranom walk. In Section 6 we iscuss the long-time near-conservation of angular momentum for systems with a rotational symmetry, which hols for constant B, but not in general. While we explain here that the Boris metho, espite some remarkable properties, is not fully satisfactory with regar to energy an momentum conservation (unless the magnetic fiel is constant), we mention that recently several classes of explicit methos for (1.1) were evelope that have longtime near-conservation of energy an momentum also for general non-constant magnetic fiels [3,7,1,11]. 2 Backwar error analysis The iea of backwar error analysis consists of searching for a moifie ifferential equation (as a formal series in powers of h) such that its solution y(t) formally satisfies y(nh) = x n, where x n represents the numerical solution obtaine by the Boris algorithm. Such a function has to satisfy y(t + h) 2y(t) + y(t h) y(t + h) y(t h) h 2 = B ( y(t) ) U ( y(t) ). 2h Expaning all appearing functions into powers of h yiels (omitting the obvious argument t) ÿ + h2... y +... = 12 (ẏ + h2... ) y +... B(y) U(y). (2.1) 6 For h =, the equation reuces to (1.1). The thir an higher erivatives can be recursively eliminate by repeately ifferentiating the equation an setting h to. This then gives the moifie ifferential equation, which is a secon-orer ifferential equation whose right-han sie is a formal series in even powers of the step size h with coefficient functions that epen on (y, ẏ): ÿ = ẏ B(y) U(y) + h 2 F 2 (y, ẏ) + h 4 F 4 (y, ẏ) To get a system of first orer moifie ifferential equations that is consistent with (1.4) we introuce the velocity approximation w(t) by y(t + h) y(t h) = 2h w(t), so that formally w(nh) = v n. Expaning into powers of h yiels the relation w = ẏ + h2... y + h4 3! 5! y(5) (2.2) The thir an higher erivatives of y can be expresse with the help of (2.1) in terms of (y, ẏ). This then permits us to express ẏ in terms of (y, w).

4 4 E. Hairer, Ch. Lubich Taking the inner prouct of (2.1) with ẏ or with w, we will prove in the next two sections that the Boris metho nearly preserves the energy over very long times if B is constant or if U is quaratic, as state in the following theorem. Theorem 2.1 Let the magnetic fiel B(x) an the scalar potential U(x) be arbitrarily ifferentiable functions of x, an suppose that the numerical solution (x n, v n ) of the Boris metho stays in a compact set that is inepenent of the step size h. In the case that one of the following two conitions is satisfie: the magnetic fiel B(x) = B is constant, or the scalar potential U(x) = 1 2 x Qx + q x is quaratic, then for arbitrary truncation number N 2, the eviation of the energy E(x, v) = 1 2 v v + U(x) along the numerical solution is boune by E(x n, v n ) E(x, v ) C N h 2 for nh h N, (2.3) where C N is inepenent of n an h as long as nh h N. Remark 2.1 If the level sets of the energy E(x, v) are compact, then the energy boun ensures, via an inuction argument, that the numerical solution of the Boris metho stays in a fixe compact set over times nh h N : if (2.3) hols true for n, then the numerical solution at step n + 1 has a istance to the energy surface of at most O(h) an hence stays in a compact set, so that by Theorem 2.1 the boun (2.3) hols also for n + 1 as long as (n + 1)h h N. 3 Moifie energy constant magnetic fiel To prove near-conservation of energy for the Störmer Verlet metho (in the absence of the magnetic fiel), one takes the scalar prouct of (2.1) with ẏ an thus fins a moifie energy that is close to E(x, v) [4]. In the present situation, this proceure yiels the following result. Theorem 3.1 There exist h-inepenent functions E 2j (x, v) such that the function E h (x, v) = E(x, v) + h 2 E 2 (x, v) + h 4 E 4 (x, v) +..., truncate at the O(h N ) term, satisfies t E h(y, ẏ) = h 2 ẏ (( 1... ) ) y + h2 3! 5! y(5) +... B(y) + O(h N ) (3.1) along solutions of the moifie ifferential equation (2.1). Proof Multiplie with ẏ T, even-orer erivatives of y give a total erivative: ẏ T y (2m) = t ( ẏ T y (2m 1) ÿ T y (2m 2) +... (y (m 1) ) T y (m+1) ± 1 2 (y(m) ) T y (m)).

5 Energy behaviour of the Boris metho for charge-particle ynamics 5 Multiplication of (2.1) with ẏ therefore yiels ( 1 2ẏ ẏ + U(y) + h2 (ẏ... y 1 2ÿ ÿ ) ) +... = h2 t 12 6 ẏ (... ) y B(y) The fact that ẏ is orthogonal to ẏ B(y) is use on the right-han sie. Expressing secon an higher erivatives of y with the help of the moifie ifferential equation proves the statement of the theorem. Corollary 3.1 If the magnetic fiel B(x) = B is constant, there exist h- inepenent functions Ẽ2j(x, v), such that the function Ẽ h (x, v) = E(x, v) + h 2 Ẽ 2 (x, v) + h 4 Ẽ 4 (x, v) +..., truncate at the O(h N ) term, satisfies tẽh(y, ẏ) = O(h N ) (3.2) along solutions of the moifie ifferential equation (2.1). Proof This follows from Theorem 3.1, because the expression ẏ ( y (k) B ) can be written as a total erivative for o values k = 2m + 1, namely as ( ẏ ( y (2m) B ) ÿ ( y (2m 1) B ) + ± (y (m 1) ) ( y (m+1) B )), t since (y (m) ) ( y (m) B ) =. This total erivative can be move to the other sie of (3.1) to obtain (3.2). If the numerical solution stays in a compact set that is inepenent of h, then Corollary 3.1 implies the long-time near-conservation of energy (2.3) by the stanar argument of writing the eviation of the moifie energy as a telescoping sum, Ẽ h (x n, v n ) Ẽh(x, v ) = n (Ẽh (x j, v j ) Ẽh(x j 1, v j 1 ) ), j=1 where each term in the sum is of size O(h N+1 ), uniformly for (x j, v j ) in the compact set. Alternatively to this proof of long-time near-conservation of energy in the case of constant B, (2.3) follows also from the known theory of symplectic integrators, e.g. [5, Chap. IX], since the Boris metho is known to be symplectic in the case of constant B.

6 6 E. Hairer, Ch. Lubich 4 Moifie energy quaratic electric potential Instea of ẏ, the equation (2.1) is now pre-multiplie with the expression in front of B(y). This implies that the term with the magnetic fiel isappears. Theorem 4.1 There exist h-inepenent functions E 2j (x, v) (ifferent from those of the previous section), such that the function E h (x, v) = E(x, v) + h 2 E 2 (x, v) + h 4 E 4 (x, v) +..., truncate at the O(h N ) term, satisfies t E h(y, ẏ) = h 2( 1... ) y + h2 3! 5! y(5) +... U(y) + O(h N ) (4.1) along solutions of the moifie ifferential equation (2.1). Proof The proof is similar to that of Theorem 3.1. One uses the fact that the scalar prouct y (k) y (l) is a total ifferential whenever k + l is o. Corollary 4.1 If the scalar potential is quaratic, U(x) = 1 2 x Qx+q x with a symmetric matrix Q, then there exist h-inepenent functions Ê2j(x, v) such that the function Ê h (x, v) = E(x, v) + h 2 Ê 2 (x, v) + h 4 Ê 4 (x, v) +..., truncate at the O(h N ) term, satisfies têh(y, ẏ) = O(h N ) (4.2) along solutions of the moifie ifferential equation (2.1). Proof This follows from Theorem 4.1, because the expression y (k) U(y) = y (k) (Qy + q) can be written as a total ifferential for o values of k. This result shows that for a quaratic potential U(x) = 1 2 x Qx + q x the energy is nearly preserve inepenently of the form of the magnetic fiel B(x), with the same relation as in (2.3). An even stronger result hols. Theorem 4.2 Let the scalar potential be quaratic, U(x) = 1 2 x Qx+q x with a symmetric matrix Q. Then, for every vector fiel B(x), the Boris metho (1.4) exactly conserves the iscrete moifie energy in the sense that E h (x, v) = 1 2 vt v + U(x) h 2 vt U(x) E h (x n+1, v n+1/2 ) = E h (x n, v n 1/2 ).

7 Energy behaviour of the Boris metho for charge-particle ynamics 7 Proof Taking the scalar prouct of the lower relation of (1.4) with v n = (v n+1/2 + v n 1/2 )/2 yiels 1 2 v n+1/2 v n+1/2 1 2 v n 1/2 v n 1/2 = h 2 (v n+1/2 + v n 1/2 ) (Qx n + q). Aing the ientity U(x n+1 ) U(x n ) = 1 2 x n+1qx n x n Qx n + q (x n+1 x n ) 1 ) = hvn+1/2( 2 Q (x n+1 + x n ) + q an observing that the terms with hvn+1/2 Qx n cancel, this then proves the statement of the theorem. Remark 4.1 An interesting example (penning trap with asymmetric magnetic fiel), for which the Boris algorithm shows an unboune energy, is given in [8] (see also [7]) by ( ) U(x) = 5 x x x 2 3, B(x) = 1 1/3 x 2 x x 1 + x 3 x 2 x 1 with initial values x() = ( 1/3,, 1/2 ), v() = (, 1, ). Since the potential U(x) is quaratic, Theorem 4.2 applies. This shows that the energy can be unboune only if the numerical solution oes not stay in a compact set. Note that the level sets of the energy E(x, v) are not compact for this example. 5 Numerical experiments This section stuies the long-time behaviour of the numerical energy in the situation where the electric potential is non-quaratic an the magnetic fiel is non-constant. Inspire by the work [6], where a symmetric, non-symplectic, but volume-preserving integrator for molecular ynamics simulations is stuie, we expect that the following two situations can arise: a linear energy-rift of size O(h 2 t); a ranom walk behaviour for the error of the numerical energy. Before passing to the numerical experiments we write the moifie energies of Sections 3 an 4 in terms of y an w. We then have formally x n = y(nh) an v n = w(nh), where (x n, v n ) is the numerical solution obtaine by (1.4). For this we use the relation (2.2) connecting w with ẏ. We have E(y, ẏ) = E(y, w) + O(h 2 ). As a consequence of Theorems 3.1 an 4.1 the solution ( y(t), w(t) ) of the moifie ifferential equation is seen to satisfy ( ) E(y, w) + h 2 F (y, w) = h 2 G(y, w) + O(h 4 ). (5.1) t

8 8 E. Hairer, Ch. Lubich h 2 step size h =.1 Ten = 3 h 2 step size h =.2 Ten = 3 Fig. 5.1 Energy error E(x n, v n) E(x, v ) along numerical solution, for Example 5.1 In the following numerical experiments we plot E(x n, v n ). For both examples of this section we consier the scalar potential U(x) = x 3 1 x x4 1 + x x 4 3 (5.2) an initial values x() = (., 1.,.1), v() = (.9,.55..3). (5.3) The quartic terms imply that the level sets of the energy are compact, so that the exact solution of the problem exists an remains boune for all times. Example 5.1 (Linear rift in the energy) In aition to the scalar potential (5.2) an the initial values (5.3) we consier the magnetic fiel B(x) = 1 3 x 2 x x 2 2 x 1 x x 2 2 = x x 2 2. (5.4) We apply the Boris algorithm in its stabilize form (1.4) with step sizes h =.1 an h =.2. The error in the energy E(x n, v n ) E(x, v ) along the numerical solution is plotte in Figure 5.1 as a function of time. The vertical axis is scale by h 2. Since the figures for both step sizes are similar, we conclue that there is a linear rift of slope O(h 2 ) in the numerical energy, which is superpose by high oscillations of size O(h 2 ). This rift in the energy is somewhat expecte. Integrating the relation (5.1) shows that, in general, we have a linear rift boune by th 2 M, where M is an upper boun of G(y, w). The term h 2 F (y, w) gives rise to boune, high oscillations of size O(h 2 ).

9 Energy behaviour of the Boris metho for charge-particle ynamics h 2 step size h =.1 Ten = 3 h 2 step size h =.2 Ten = 3 Fig. 5.2 Energy error E(x n, v n) E(x, v ) along numerical solution, for Example 5.2 Example 5.2 (Ranom walk behaviour of the numerical energy) This time we consier the magnetic fiel B(x) = 1 x 2 3 x 2 2 x 2 3 x 2 1 = 1 x 2 x 3 x 1 + x 3. (5.5) 4 2 x 2 2 x 2 1 x 2 x 1 in aition to (5.2) an (5.3). We apply the Boris algorithm (1.4) with step sizes h =.1 an h =.2 on an interval of length T en = 3. To better appreciate the ranom walk behaviour, we compute the numerical solution for initial values, where v 3 () is replace by v 3 () + θ1 13. Here, θ is ranomly chosen in the interval [, 1]. Figure 5.2 shows the energy error of 41 trajectories. Thick lines inicate average an variance taken over 11 trajectories. This behaviour can be explaine as in [6]. Let us assume that the solution of the moifie ifferential equation is ergoic on an invariant set A with respect to an invariant measure µ. Then we have 1 lim t t t G ( y(s), w(s) ) s = A G(x, v) µ ( (x, v) ) for the function G(x, v) of (5.1). If the integral to the right is non-zero, we will have a linear rift of size O(th 2 ). However, if this integral vanishes, the error of the moifie energy will behave like a ranom walk. In aition to the O(h 2 ) term h 2 F (x, v) of (5.1) there will be a rift of size t h 2. 6 Momentum If the scalar an vector potentials have the invariance properties U(e τs x) = U(x) an e τs A(e τs x) = A(x) for all real τ (6.1)

10 1 E. Hairer, Ch. Lubich with a skew-symmetric matrix S, then the momentum M(x, v) = ( v + A(x) ) Sx (6.2) is conserve along solutions of the ifferential equation (1.1). This can be shown by scalarly multiplying (1.1) with Sx an noting that the skew-symmetry of S yiels x Sẍ = t (xt Sẋ) an the invariance properties (6.1) imply S U(x) = an x S(ẋ B(x)) = t (x SA(x)). Theorem 6.1 If the vector an scalar potentials have the invariance properties (6.1), then there exist h-inepenent functions M 2j (x, v) such that the function M h (x, v) = M(x, v) + h 2 M 2 (x, v) + h 4 M 4 (x, v) +..., truncate at the O(h N ) term, satisfies (( 1 t M h(y, ẏ) = h 2 y... ) ) S y + h2 3! 5! y(5) +... B(y) + O(h N ) (6.3) along solutions of the moifie ifferential equation (2.1). Proof We multiply (2.1) with y S an note that y S y (k) can be written as a total ifferential for even values of k, an y S U(y) =. This yiels the state result in the same way as in Theorem 3.1. When B(x) = B oes not epen on x, then A(x) = 1 2x B (up to a constant vector) an the above invariance properties are satisfie if S is the skew-symmetric matrix that emboies the cross prouct with B, i.e., Sv = v B, an U is invariant uner rotations with the axis B, so that U(x) B = for all x. The conserve momentum then reas M(x, v) = v (x B) 1 2 x B 2. Corollary 6.1 If the magnetic fiel B(x) = B is constant an U(x) B = for all x, then there exist h-inepenent functions M 2j (x, v), such that the function M h (x, v) = M(x, v) + h 2 M2 (x, v) + h 4 M4 (x, v) +..., truncate at the O(h N ) term, satisfies t M h (y, ẏ) = O(h N ) (6.4) along solutions of the moifie ifferential equation (2.1). Proof This follows from Theorem 6.1, since for constant B we have with Sy = y B that (y B) (y (k) B) can be written as a total ifferential for o values of k.

11 Energy behaviour of the Boris metho for charge-particle ynamics 11 If the numerical solution stays in a compact set that is inepenent of h, then Corollary 3.1 implies, for arbitrary positive integers N, that M(x n, v n ) = M(x, v ) + O(h 2 ) for nh Ch N, (6.5) where the constant symbolize by the O-notation is inepenent of n an h with nh Ch N. Acknowlegement. We thank our colleague Martin Ganer for rawing our attention to the long-time behaviour of the Boris algorithm. The research for this article has been partially supporte by the Fons National Suisse, Project No References 1. J. P. Boris. Relativistic plasma simulation-optimization of a hybri coe. Proceeing of Fourth Conference on Numerical Simulations of Plasmas, pages 3 67, November C. L. Ellison, J. W. Burby, an H. Qin. Comment on Symplectic integration of magnetic systems : A proof that the Boris algorithm is not variational. J. Comput. Phys., 31: , E. Hairer an C. Lubich. Symmetric multistep methos for charge particle ynamics. SMAI J. Comput. Math., 3:25 218, E. Hairer, C. Lubich, an G. Wanner. Geometric numerical integration illustrate by the Störmer Verlet metho. Acta Numerica, 12:399 45, E. Hairer, C. Lubich, an G. Wanner. Geometric Numerical Integration. Structure- Preserving Algorithms for Orinary Differential Equations. Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin, 2n eition, E. Hairer, R.I. McLachlan, an R.D. Skeel. On energy conservation of the simplifie Takahashi Imaa metho. M2AN Math. Moel. Numer. Anal., 43(4): , Yang He, Zhaoqi Zhou, Yajuan Sun, Jian Liu, an Hong Qin. Explicit K-symplectic algorithms for charge particle ynamics. Phys. Lett. A, 381(6): , C. Knapp, A. Kenl, A. Koskela, an A. Ostermann. Splitting methos for time integration of trajectories in combine electric an magnetic fiels. Phys. Rev. E, 92:6331, Dec H. Qin, S. Zhang, J. Xiao, J. Liu, Y. Sun, an W. M. Tang. Why is Boris algorithm so goo? Physics of Plasmas, 2(8): , M. Tao. Explicit high-orer symplectic integrators for charge particles in general electromagnetic fiels. J. Comput. Phys., 327: , Ruili Zhang, Hong Qin, Yifa Tang, Jian Liu, Yang He, an Jianyuan Xiao. Explicit symplectic algorithms base on generating functions for charge particle ynamics. Physical Review E, 94(1):1325, 216.