Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations

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1 Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations D. COHEN 1, E. HAIRER 2 and CH. LUBICH 3 April 28, Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway. David.Cohen@math.ntnu.no 2 Dept. de Mathématiques, Univ. de Genève, CH-1211 Genève 4, Switzerland. Ernst.Hairer@math.unige.ch 3 Mathematisches Institut, Univ. Tübingen, D Tübingen, Germany. Lubich@na.uni-tuebingen.de Abstract For classes of symplectic and symmetric time-stepping methods trigonometric integrators and the Störmer Verlet or leapfrog method applied to spectral semi-discretizations of semilinear wave equations in a weakly nonlinear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time. sect:intro 1 Introduction This paper is concerned with the long-time behaviour of symplectic integrators applied to Hamiltonian nonlinear partial differential equations, such as semilinear wave equations. For symplectic methods applied to Hamiltonian systems of ordinary differential equations, the numerically observed long-time near-conservation of the total energy, and of actions in near-integrable systems, can be rigorously explained with the help of backward error analysis. This interprets a step of a symplectic method as the exact flow of a modified Hamiltonian system, up to an error which in the case of an analytic Hamiltonian is exponentially small in 1/hω, where h is the small step size and ω represents the largest frequency in a local linearization of the system; see Benettin & Giorgilli [2], Hairer & Lubich [11], Reich [16], and Chapter IX in Hairer, Lubich & Wanner [14]. When the symplectic method is applied to a semi-discretization of a partial differential equation, however, then the product hω corresponds to the CFL number, which in typical computations is not small but of size 1. In this situation, the exponentially small remainder terms become of magnitude O1, and no conclusions on the long-time behaviour of the method can then be drawn from the familiar backward error analysis. Nevertheless, long-time conservation of energy, and of momentum and actions when appropriate, is observed in numerical computations with symplectic methods used with reasonable CFL 1

2 numbers. The present paper gives a theoretical explanation of such conservation properties in the case of semilinear wave equations in the weakly nonlinear regime, over time scales that go far beyond linear perturbation arguments. To our knowledge, the results of this paper are the first results that rigorously explain the remarkable long-time conservation properties of symplectic integrators on a class of nonlinear partial differential equations. We consider the one-dimensional nonlinear wave equation u tt u xx + ρu + gu = 0 1 nlw for t > 0 and π x π subect to periodic boundary conditions. We assume ρ > 0 and a nonlinearity g that is a smooth real function with g0 = g 0 = 0. We consider small initial data: in appropriate Sobolev norms, the initial values u, 0 and u t, 0 are bounded by a small parameter ε. In Section 2 we recall the exact conservation of energy and momentum and, less obvious, the near-conservation of actions over long times t ε N, where N only depends on a non-resonance condition on the frequencies, as shown by Bambusi [1] and Bourgain [3]. With the technique of modulated Fourier expansions that is central also to the present paper, the near-conservation of actions along solutions of 1 has been studied in our paper [6], and for spatial semi-discretizations of 1 by spectral methods in [13]. After discussing the semi-discretization in Section 3, we turn to the time discretization in Section 4. We consider a class of symplectic and symmetric trigonometric integrators discussed in [14, Chap. XIII], and the familiar Störmer Verlet or leapfrog method. In Section 4 we describe the trigonometric methods and present numerical experiments illustrating their conservation properties, which appear particularly remarkable when confronted with the behaviour of a standard explicit Runge-Kutta method. In Section 5 we state the main result of this paper, concerning the long-time near-conservation of energy, momentum and actions along numerical solutions in the full discretization. The result is proved in Sections 6 and 7, using the technique of modulated Fourier expansions. This approach was first used for studying long-time conservation properties of numerical methods for highly oscillatory ordinary differential equations with a single high frequency in [12], and later extended to several frequencies in [5]; see also [14, Chap. XIII] and further references given there. The extension of this technique to infinitely many frequencies, as occur in equation 1, was studied for the analytical problem in [6], and our treatment here essentially follows the lines of this previous work, with additional technical complications arising from the discretization. In Section 8 we give similar long-time conservation results for the Störmer Verlet/leapfrog method used with step sizes in the linear stability interval. These results follow from the previous ones by interpreting the leapfrog method as a trigonometric method with modified frequencies. We are aware of two other papers that deal with long-time energy conservation of symplectic integrators for partial differential equations. Cano [4] also considers the nonlinear wave equation and aims at extending the classical backward error analysis to this situation. Long-time conservation properties are obtained under a list of unverified conditions formulated as conectures. For symplectic splitting methods applied to the linear Schrödinger equation with a small potential, results on long-time energy conservation are given by Duardin & Faou [8]. 2

3 sect:waveeq 2 The nonlinear wave equation with small data The semilinear wave equation 1 conserves several quantities along every solution ux, t, vx, t, with v = t u. The total energy or Hamiltonian, defined for 2π-periodic functions u, v as Hu, v = 1 2π π π 1 2 v 2 + x u 2 + ρ u 2 x + U ux dx, 2 hamilanalyt where the potential Uu is such that U u = gu, and the momentum Ku, v = 1 π x uxvxdx = i u v 2π 3 momentumalyt π = are exactly conserved along every solution u, t, v, t of 1. Here, u = F u and v = F v are the Fourier coefficients in the series ux = = u e ix and correspondingly vx. Since we consider only real solutions, we note that u = u and v = v. In terms of the Fourier coefficients, equation 1 reads with the frequencies The harmonic actions 2 t u + ω 2 u + F gu = 0, Z, 4 nlw ω = ρ + 2. I u, v = ω 2 u ω v 2, 5 actions for which we note I = I, are conserved for the linear wave equation, that is, for gu 0. In the semilinear equation 1, they turn out to remain constant up to small deviations over long times for almost all values of ρ > 0, when the initial data are smooth and small [1, 3, 6]. We recall the precise statement of this result, because this will help to understand related assumptions for the numerical discretizations. We work with the Sobolev space, for s 0, H s = {v L 2 T : v s < }, v s = = ω 2s v 2 1/2, where v denote the Fourier coefficients of a 2π-periodic function v. For the initial position and velocity we assume that for suitably large s and small ε, u, 0 2 s+1 + v, 0 2 s 1/2 ε. 6 small-init Since the analysis of the near-conservation of actions encounters problems with small denominators, we prepare for the formulation of a non-resonance condition. Consider sequences of integers k = k l l=0 with only finitely many k l 0. We denote k = k l l=0 and let k = k l, k ω = k l ω l, ω σ k = ω σ k l l 7 prod-omegas l=0 l=0 for real σ, where we use the notation ω = ω l l=0. For Z, we write = 0,..., 0, 1, 0,... with the only entry at the -th position. l=0 3

4 For an arbitrary fixed integer N and for small ε > 0, we consider the set of near-resonant indices Theorem 2.1 [6, Theorem 1] Under the non-resonance condition 9 and as- sumption 6 on the initial data with s σ + 1, the estimate ω 2s+1 I l t I l 0 l ε 2 Cε for 0 t ε N+1 thm:conserve sect:semi-disc R ε = {,k : Z and k ±, k 2N with ω k ω < ε 1/2 }. 8 We impose the following non-resonance condition: there are σ > 0 and a constant C 0 such that sup,k R ε res-seta ωσ ω σ k ε k /2 C 0 ε N. 9 nonresanal As is shown in [6], condition 9 is implied, for sufficiently large σ, by the non-resonance condition of Bambusi [1], which holds true for almost all w.r.t. Lebesgue measure ρ in a fixed interval of positive numbers. l=0 with I l t = I l u, t, v, t holds with a constant C which depends on s, N, and C 0, but is independent of ε and t. 3 Spectral semi-discretization in space For the numerical solution of 1 we first discretize in space method of lines and then in time Section 4. Following [13], we consider pseudo-spectral semidiscretization in space with equidistant collocation points x k = kπ/m for k = M,...,M 1. This yields an approximation in form of real-valued trigonometric polynomials u M x, t = q te ix, v M x, t = p te ix 10 umvm M M where the prime indicates that the first and last terms in the sum are taken with the factor 1/2. We have p t = d dt q t, and the 2M-periodic coefficient vector qt = q t is a solution of the 2M-dimensional system of ordinary differential equations d 2 q dt 2 + Ω2 q = fq with fq = F 2M gf 1 2Mq. 11 nlw-semidisc The matrix Ω is diagonal with entries ω for M, and F 2M denotes the discrete Fourier transform: F 2M w = 1 M 1 2M k= M w k e ix k. Since the components of the nonlinearity in 11 are of the form f q = V q with V q = 1 q 2M M 1 k= M U F 1 2M q k, we are concerned with a finite-dimensional complex Hamiltonian system with energy H M q, p = 1 p 2 + ω 2 q 2 + V q, 12 hamil-semi 2 M 4

5 which is exactly conserved along the solution qt, pt of 11 with pt = dqt/dt. We further consider the actions for M and the momentum I q, p = ω 2 q ω p 2, Kq, p = i q p, 13 momentum-semi M where the double prime indicates that the first and last terms in the sum are taken with the factor 1/4. The definition of these expressions is motivated by the fact that they agree with the corresponding quantities of Section 2 along the trigonometric polynomials u M, v M with the exception of I ±M, where a factor 4 has been included to get a unified formula. Since we are concerned with real approximations 10, the Fourier coefficients satisfy q = q and p = p, so that also I = I. On the space of 2M-periodic sequences q = q we consider the weighted norm q s = ω 2s q 2 1/2, 14 normdisc M which is defined such that it equals the H s norm of the trigonometric polynomial with coefficients q. We assume that the initial data q0 and p0 satisfy a condition corresponding to 6: q0 2 s+1 + p0 2 s 1/2 ε. 15 initial thm:conserve-semii Theorem 3.1 [13, Theorems 3.1 and 3.2] Under the non-resonance condition 9 with exponent σ and the assumption 15 of small initial data with s σ + 1, the near-conservation estimates M l=0 ω 2s+1 l I l t I l 0 ε 2 Kt K0 ε 2 Cε C t εm s 1 for 0 t ε N+1 for actions I l t = I l qt, pt and momentum Kt = K qt, pt hold with a constant C that depends on s, N, and C 0, but is independent of ε, M, and t. Since the expression M l=0 ω2s+1 l I l t is essentially up to the factors in the boundary terms equal to the squared H s+1 H s norm of the solution qt, pt, Theorem 3.1 implies long-time spatial regularity: qt 2 s+1 + pt 2 s 1/2 ε1 + Cε for t ε N regularity Theorems 2.1 and 3.1 have been included as a motivation of our results. They will not be used in the following. sect:numer 4 Full discretization and numerical phenomena We consider the class of time discretization methods studied in [14, Chapter XIII], which gives the exact solution for linear problems 11 with fq = 0, and reduces to the Störmer Verlet/leapfrog method for 11 with Ω = 0: q n+1 2 coshωq n + q n 1 = h 2 Ψ fφq n, 17 gautschi 5

6 where Ψ = ψhω and Φ = φhω with filter functions ψ and φ that are realvalued, bounded, even, and satisfy ψ0 = φ0 = 1. A velocity approximation p n is obtained from 2hsinc hωp n = q n+1 q n 1 18 gautschi-velocity provided that sinchω is invertible. Here we use the notation sinc ξ = sinξ/ξ. For an implementation it is more convenient to work with an equivalent one-step mapping q n, p n q n+1, p n+1, which is obtained from adding and subtracting the formulas 17 and 18 and which reads q n+1 = coshωq n + hsinc hωp n h2 ΨfΦq n p n+1 = Ω sinhωq n + coshωp n + 1 Ψ h 0 fφq n + Ψ 1 fφq n one-step Here, Ψ 0 = ψ 0 hω and Ψ 1 = ψ 1 hω, where the functions ψ i ξ are defined by the relations ψξ = sincξψ 1 ξ and ψ 0 ξ = cosξψ 1 ξ. These methods are symmetric for all choices of ψ and φ; they are symplectic if ψξ = sinc ξ φξ for all real ξ. 20 symplectic The methods 19 with this property are precisely the mollified impulse methods introduced in [9]. Condition 20 will be assumed in the following. We note, however, that for non-symplectic methods, the transformation of variables q n = χhωq n, p n = χhωp n, 21 transf turns the method 19 into a symplectic method if χ can be chosen as a positive solution of χξ 2 = φξ sinc ξ/ψξ. In our numerical experiments we consider the nonlinear wave equation 1 with the following data: ρ = 0.5, gu = u 2, and initial data x 3 x 2, ux, 0 = 0.1 π 1 π + 1 t ux, 0 = 0.01 x x x 2 π π 1 π + 1 for π x π. The spatial discretization is 11 with dimension 2M = 2 7. We first apply a standard explicit Runge Kutta method in the variable stepsize implementation DOPRI5 of [15], with local error tolerances Atol = 10 5 and Rtol = The program chose accepted steps for the integration over the interval 0 t 550, which corresponds to an average stepsize h = and average CFL number hω M = In both pictures of Figure 1 we plot the actions I of 5, the total energy H M of 12, and the momentum K of 13 along the numerical solution. The left-hand picture illustrates that even on the short interval 0 t 1, the actions with values below the tolerance are not at all conserved. The right-hand picture shows substantial drifts in all the quantities over a longer time interval. We now consider method 19 with ψ = sinc and φ = 1, which was originally proposed in [7]. The method can also be viewed as a special case of the impulse method used in molecular dynamics [10, 17]. We apply the method with stepsize h = 0.1 to the above problem. The CFL number then is hω M 6.4. Figure 2 illustrates that energy, momentum and actions are very well conserved. In a further experiment with the same problem, we choose stepsizes such that hω 4 is close to π Figure 3. In this situation of a numerical resonance, 6

7 Figure 1: Actions, total energy upper bold line, and momentum lower bold line along the numerical solution of DOPRI5, average CFL number fig:sine_dopri Figure 2: Actions, total energy upper bold line, and momentum lower bold line along the numerical solution of the trigonometric integrator 19 with ψ = sinc and φ = 1 for the CFL number hω M 6.4. fig:driver_sine the action I 4 is no longer preserved, which on longer time scales also affects the conservation of energy. The resonance behaviour depends strongly on the choice of the filter functions, cf. [14, Section XIII.2]. For example, with φ = sinc and ψ = sinc 2, no numerical resonance is visible. 7

8 10 2 hω 4 = π 0.01 hω 4 = π hω 4 = π Figure 3: Illustration of numerical resonance. fig:impulse_wave_reson We now turn to a theoretical explanation of the observed numerical conservation properties. sect:main 5 Main results To explain the good long-time behaviour illustrated in Section 4, we combine the techniques of [6], where the long-time preservation of the harmonic actions along exact solutions of the semilinear wave equation 1 is shown, with those of [5], where the long-time behaviour of the numerical method 17 is studied for oscillatory Hamiltonian systems with a fixed number of large frequencies. As for spectral semi-discretizations cf. [13], we are interested in results that are valid uniformly in M, where 2M is the dimension of the spatially discretized system 11. The analytical tool for understanding the long-time behaviour of the numerical solution of 11 is given by a modulated Fourier expansion in time see [14, Chapter XIII] and [6], qt = z k εte ik ωt, 22 mfe k 2N approximating the numerical solution q n at t = nh. We use the notation introduced in 7, where now k l = 0 for l > M, since only the frequencies ω l for 0 l M appear in the spatial discretization 11. In our analysis, we must deal with small denominators see Section 6. To control these terms, we will use non-resonance conditions. As soon as, for a given step size h, the inequality h h sin 2 ω k ω sin 2 ω + k ω ε 1/2 h 2 ω + k ω 23 res-cond 8

9 is violated, we have to make an assumption on the pair of indices,k. For a fixed integer N 1, subsequently used in the truncation of the expansion 22, the set of near-resonant indices becomes, instead of 8, R ε,h = {,k : M, k 2N, k ±, not satisfying 23 }. 24 res-set Similar to 9, we require the following non-resonance condition: there are σ > 0 and a constant C 0 such that sup,k R ε,h ωσ ω σ k ε k /2 C 0 ε N. 25 nonres Notice that, in the limit h 0, condition 23 becomes equivalent up to a non-zero constant factor to ω 2 k ω 2 ε 1/2 ω + k ω, so that 25 corresponds precisely to the non-resonance condition 9 for the semilinear wave equation. We assume the further numerical non-resonance condition sinhω hε 1/2 for M. 26 nores2 Yet another non-resonance condition, which leads to improved conservation estimates, reads as follows: h h sin 2 ω k ω sin 2 ω + k ω c h 2 ψhω for,k of the form = and k = ± 1 ± 2, with a positive constant c > 0. In the limit h 0, this inequality becomes ω 2 k ω 2 4c which is automatically fulfilled for the considered pairs,k. We are now in the position to state the main result of this paper. 27 nores3 thm:conserve-full Theorem 5.1 Under the symplecticity condition 20, under the non-resonance conditions 25 with exponent σ and 26-27, and under the assumption 15 of small initial data with s σ+1 for q 0, p 0 = q0, p0, the near-conservation estimates M l=0 H M q n, p n H M q 0, p 0 ε 2 Cε Kq n, p n Kq 0, p 0 ε 2 C ε + M s + εtm s+1 ω 2s+1 l I l q n, p n I l q 0, p 0 ε 2 Cε for energy, momentum and actions hold for long times 0 t = nh ε N+1 with a constant C which depends on s, N, and C 0, but is independent of the small parameter ε, the dimension 2M of the spatial discretization, the time stepsize h, and the time t = nh. If condition 27 fails to be satisfied, then Cε is weakened to Cε 1/2 in the above bounds. 9

10 In addition we obtain, by the argument of Section 6.2 in [13], that the original Hamiltonian H of 2 along the trigonometric interpolation polynomials u n x, v n x with Fourier coefficients q n, pn satisfies the long-time nearconservation estimate Hu n, v n Hu 0, v 0 ε 2 Cε for 0 nh ε N+1. For a non-symplectic symmetric method 19 the result remains valid in the transformed variables 21. The proof of Theorem 5.1 is given in the subsequent Sections 6 and 7. sect:modfourier 6 Modulated Fourier expansion Our principal tool for the long-time analysis of the nonlinearly perturbed wave equation is a short-time modulation expansion constructed in this section. To construct this expansion, we combine the tools and techniques developed in [5], [6], and [13]. 6.1 Statement of the result In this section we consider, instead of the symplecticity condition 20, the weaker condition ψhω C sinc hω for M. 28 psi-cond thm:mfe In the following result we use the abbreviations 7 and set [[k]] = 1 k + 1, 2 k k = 0. Theorem 6.1 Under the assumptions of Theorem 5.1 with the symplecticity assumption 20 relaxed to 28, there exist truncated asymptotic expansions with N from 25 qt = k 2N z k εte ik ωt 1 qt + h qt h, pt = sinc hω, 29 uhvh 2h such that the numerical solution q n, p n given by method 19, satisfies q n qt s+1 + p n pt s Cε N for 0 t = nh ε mfe-err The truncated modulated Fourier expansion is bounded by qt s+1 + pt s Cε for 0 t ε mfe-bound-qp On this time interval, we further have, for M, q t = z εte iωt + z εte iωt + r, with r s+1 Cε mfe-r 10

11 If condition 27 fails to be satisfied, then the bound is r s+1 Cε 3/2. The modulation functions z k are bounded by ω k 2 ε [[k]] zk εt s C. 33 zk-bound k 2N Bounds of the same type hold for any fixed number of derivatives of z k with respect to the slow time τ = εt. Moreover, the modulation functions satisfy z k = zk. The constants C are independent of ε, M, h, and of t ε 1. The proof of this result will cover the remainder of this section. It is organized in the same way as the proof of the analogous result for the analytical solution in [6]. 6.2 Formal modulation equations We are looking for a truncated series 29 such that, up to a small defect, qt + h 2 coshω qt + qt h = h 2 Ψ fφ qt with q0 = q 0, p0 = p 0, see 17 and 29. We insert the ansatz 29 into this equation, expand the right-hand side into a Taylor series around zero and compare the coefficients of e ik ωt. We then get L k zk = h 2 ψhω m 2 g m 0 m! k k m =k m mod2m φhω 1 z k φhω m z km m, 34 diff-l2 where the right-hand side is obtained as in [13]. The prime on the sum over 1,..., m indicates that with every appearance of z ki i with i = ±M a factor 1/2 is included. The operator L k is given as L k z k τ = e ihk ω z k τ + εh 2 coshω z k τ + e ihk ω z k τ εh = 4s +k s k z k τ + 2is 2k hεż k τ + c 2k h 2 ε 2 z 35 LhD k τ Here, s k = sin h k ω and c k = cos h k ω, and the dots on 2 2 zk represent derivatives with respect to the slow time τ = εt. The higher order terms are linear combinations of the rth derivative of z k for r 3 multiplied by hr ε r and containing one of the factors s 2k or c 2k. The first term in 35 vanishes for k = ±, so that in this case the dominating term becomes ±2ih sinhω εż ± due to condition 26. For k ± the first term becomes dominant, if the inequality 23 holds. Else, it is not clear which term is dominant, but then the non-resonance condition 25 will ensure that the defect in simply setting z k 0 is of size OεN+1 in an appropriate Sobolev-type norm. In addition, the initial conditions q0 = q 0 and p0 = p 0 need to be taken care of. The condition q0 = q 0 reads z k 0 = q, 0 36 mod-init k 2N 11

12 and for p0 = p 0, we obtain from 29 1 z k εheik ωh z k εhe ik ωh = p 0. 2h sinchω 37 mod-init2 k 2N subsec:iter 6.3 Reverse Picard iteration We now turn to an iterative construction of the functions z k such that after 4N iteration steps, the defect in equations 34, 36, and 37 is of size Oε N+1 in the H s norm. The iteration procedure we employ can be viewed as a reverse Picard iteration on 34 to 37, where we keep only the dominant terms on the left-hand side. Indicating by [ ] n the nth iterate of all appearing variables z k taken within the bracket, we set for k = ± [ ] n+1= [ ±2ihεs 2 ż ± h 2 ψhω g m 0 38 revpic1 m! m 2 φhω 1 z k φhω m z km m k k m =k m mod2m ] n c 2 h 2 ± ε 2 z +... with the sines and cosines s 2 and c 2 defined after formula 35. For k ± and that are non-resonant with 23, we set ] n+1= [ 4s +k s k [z k h 2 ψhω g m 0 39 revpic2 m! m 2 φhω 1 z k φhω m z km m k k m =k m mod 2M 2is 2k hεż k + c 2kh 2 ε 2 z ] n, k +... whereas we let z k = 0 for k ± in the near-resonant set R ε,h. The dots indicate the remainder in 35, truncated after the ε N term. On the initial conditions we iterate by [ z ] n z 0 = [q 0 k ± z k 0 ] n 40 inival1 and on 37 by [ n+1 iω z 0 z 0] = p 0 1 [ z e k 2h sinc hω 0 ik ωh e ik ωh 41 inival2 k K k ± z k εh z k 0 e ik ωh z k εh z k 0 e ik ωh] n. In all the above formulas, it is tacitly assumed that k K := 2N and k i K for i = 1,...,m. In each iteration step, we thus have an initial value problem of first-order differential equations for z ± for M and algebraic equations for z k with k ±. 12

13 z ± The starting iterates n = 0 are chosen as z k τ = 0 for k ±, and τ = z ± 0 with z ± 0 determined from the above formula. For real initial data we have q 0 = q0 and p0 = p0, and we observe that the above iteration yields [ z k ] n [ ] = z k n for all iterates n and all,k and hence gives real approximations 29. subsec:rescaling 6.4 Rescaling and estimation of the nonlinear terms As in [6], we will work with the more convenient rescaling c k = ω k ε [[k]] zk, ck = c k M = ω k zk ε [[k]] considered in the space H s = H s K = {c = c k k K : c k H s } with norm c 2 s = k K ck 2 s and where the superscripts k are in the set K = {k = k l M l=0 with integers k l : k K} with K = 2N. The nonlinear function f = f k defined as f k c = ω k ε [[k]] N m=2 g m 0 m! k k m =k m mod2m ε [[k1 ]]+ +[[k m ]] ω k1 + + k m φhω 1 c k φhω m c km m expresses the nonlinearity in 34 in the rescaled variables. With the fact that H s is a normed algebra, the following bounds are obtained as in [6, Section 3.5] by exploiting the connection between the 2M-periodic sequence c k and the corresponding trigonometric polynomial cf. [13]: f k c 2 s εp c 2 s 42 f1 k K M f ± c 2 s ε3 P 1 c 2 s, 43 f1 where P and P 1 are polynomials with coefficients bounded independently of ε, h, and M. Notice that the function φ is bounded. With the different rescaling ĉ k = ωs k ε [[k]] z k, ĉk = ĉ k M = ωs k ε [[k]] z k 44 rescale-1 considered in the space H 1 = H 1 K with norm ĉ 2 1 = k K ĉk 2 1, for f k defined as f k but with ω k replaced by ω s k, we have similar bounds f k ĉ 2 1 ε P ĉ 2 1 k K M with other polynomials P and P 1. f ± ĉ 2 1 ε 3 P1 ĉ fs 13

14 subsec:ab-iter 6.5 Abstract reformulation of the iteration For c = c k Hs with c k = 0 for all k ± with,k R ε,h, we split the components of c corresponding to k = ± and k ± and collect them in a = a k Hs and b = b k Hs, respectively: a k = ck if k = ±, and 0 else b k = ck if 23 is satisfied, and 0 else. 46 abc We then have a+b = c and a 2 s + b 2 s = c 2 s. We now introduce differential operators A, B acting on functions aτ and bτ, respectively: Aa ± τ = Bb k τ = 1 c 2 h 2 ε 2 ä ± τ +... ±2ihεs 2 1 2is 2k 4s +k s hεḃk τ + c 2k h 2 ε 2 bk τ +... k for,k satisfying 23. These definitions are motivated by formulas 38 and 39, and as in these formulas, the dots represent a truncation after the ε N terms. In terms of the nonlinear function f of the preceding subsection, we introduce the functions F = F k and G = Gk with non-vanishing entries F ± a,b = 1 ±iε ψhω sinc hω f ± a + b, G k a,b = ω + k ω h2 f k a + b 4s +k s k for,k satisfying 23. Further we write Ωc k = ω + k ω c k, Ψc k = ψhω c k. In terms of a and b, the iterations 38 and 39 then become of the form ȧ n+1 = Ω 1 Fa n,b n Aa n b n+1 = Ω 1 ΨGa n,b n Bb n. 47 ab-iter By 43, condition 28 gives the bound F s Cε 1/2, whereas condition 26 yields Ψ 1 Ω 1 F s C. By 42 and 23, we have the bound G s C. These bounds hold uniformly in ε, h, M on bounded subsets of H s. Analogous bounds are obtained for the derivatives of F and G. The operators A and B are estimated as Aaτ s Bbτ s C N h l 2 ε l 3/2 d l s dτ laτ l=2 N Cε 1/2 ḃτ s + C h l 2 ε l 1/2 d l s dτ lbτ. l=2 48 ABest The bound for A is obtained with 26, that for B uses 23 and the trivial estimate s 2k = sinhk ω h k ω. 14

15 The initial value conditions 40 and 41 translate into an equation for a n+1 of the form where v has the components a n+1 0 = v + Pb n 0 + Qa + b n εh 49 ab-init v ± = ω ε 1 2 q0 i p 0. 2ω By assumption 15, v is bounded in H s. The operators P and Q are given by Pb ± 0 = ω 2εs 2 Qc ± τ = ω 4iεs 2 k ± k K For these expressions we have the bounds sinω h ± sin k ωh ε [[k]] ik ωh ε[[k]] e ω k c k τ c k 0 ω k bk 0 ik ωh ε[[k]] e c k ω k τ ck 0. Pb0 s C Ψ 1 Ωb0 s Qcεh s C ε sup Ψ 1 ċτ s εh<τ<εh with a constant C that is independent of ε, h, and M, but depends on K = 2N. For the first estimate we use sinω h±sink ωh hω + k ω, condition 28, and the Cauchy Schwarz inequality together with the bound cf. Lemma 2 of [6] ω 2 k C <. 50 omega-ineq0 k K Similarly, applying the mean value theorem to cτ yields the second estimate. The starting iterates are a 0 τ = v and b 0 τ = 0. subsec:bounds-mod 6.6 Bounds of the modulation functions In view of the non-resonance conditions 23 and 26, and using the assumption on the filter function 28, we can show by induction that the iterates a n and b n and their derivatives with respect to the slow time τ = εt are bounded in H s for 0 τ 1 and n 4N: more precisely, the 4N-th iterates a = a 4N and b = b 4N satisfy a0 s C, Ωȧτ s Cε 1/2, Ψ 1 ȧτ s C, Ψ 1 Ωbτ s C, 51 ab-bounds with a constant C independent of ε, h, M, but dependent on N. We also obtain analogous bounds for higher derivatives of a and b with respect to τ = εt. For z k = ε[[k]] ω k c k with ck = c4n = a 4N + b 4N, the bounds for a and b together yield the bound

16 These bounds imply cτ a0 s+1 C and as in [6, Section 3.7] give, using 50, the bound 31 for qt. For the function pt, defined in 29, we use z kεt+εheik ωh z kεt εhe ik ωh = z kεt2isin k ωh +r k, where by the mean value theorem r k 2εh max εh<τ<εh ż k τ. Using the condition 28, the bounds 51 yield in a similar way also the statement 31 for the function pt. Using 43 and 47 we also obtain the bound, for b = b 4N, k =1 Moreover, condition 27 ensures that M 1+ 2= k=± 1 ± 2 Ψ 1 Ωb k 2 s 1/2 Cε. ω 2s+1 b k 2 Cε. These bounds together with 51 yield 32. With the alternative scaling 44 we obtain the same bounds for τ = εt 1, â0 1 C, Ω âτ 1 Cε 1/2, Ψ 1 Ω bτ 1 C. 52 ab-bounds-2 and again k =1 Ψ 1 Ω b k 2 1 1/2 Cε. 53 b1-1 For the function âτ these statements follow at once from the fact that â k 1 = a k s. For the function bτ one has to repeat the argumentation from before, but one needs no longer take care of initial values. In addition to these bounds, we also obtain that the map B ε H s+1 H s H 1 : u0, v0 ĉ0 with B ε the ball of radius ε centered at 0 is Lipschitz continuous with a Lipschitz constant proportional to ε 1 : at t = 0, â 2 â Ω b 2 b C ε 2 u 2 u 1 2 s+1 + v 2 v 1 2 s. 54 lipschitz subsect:defects 6.7 Defects We consider the defect δt = δ t M in 17 divided by h2 ψhω : δ t = q t + h 2 coshω q t + q t h h 2 ψhω f Φ qt where f = f is given in 11 and the approximation qt = q t is given by 29 with z k = z k4n obtained after 4N iterations of the procedure in Section 6.3. We write this defect as δt = d k εte ik ωt + R N+1 qt. k NK 16

17 Here we have set d k = 1 h 2 ψhω L k zk 55 defdef + N m=2 g m 0 m! k k m =k m mod2m φhω 1 z k φhω m z km m, which is to be considered for k NK, and where we set z k = 0 for k > K = 2N. The operator L k denotes the truncation of the expansion 35 after the ε N term. The function R N+1 collects the remainder term of the Taylor expansion of f after N terms, and that due to the truncation of the series in 35 after the ε N term. Using the bound 31 for the remainder in the Taylor expansion of f and the estimates 51 for the N + 1-th derivative for z kτ, we have R N+1 q s+1 Cε N+1. We now use the bound of [6, Section 3.8] to obtain d k εte ik ωt 2 C ω k d k εt. 56 sum-square-d s k NK k NK In he following two subsections we estimate the right-hand side of 56 by Cε 2N+1. 2 s subsect:deftm 6.8 Defect in the truncated and near-resonant modes For k > K = 2N truncated modes and for,k in the set R ε,h of nearresonances we have by definition z k = 0. In both situations the defect reads d k = N m=2 g m 0 m! k k m =k m mod 2M φhω 1 z k φhω m z km m For truncated modes we write the defect as d k = ε[[k]] ω k f k, and we notice that by 51 and 42, used with NK in place of K, the bound f 2 s Cε holds. We thus have ω 2s ω k d k 2 = ω 2s f k 2 ε 2[[k]] k >K M k >K M and hence, since 2[[k]] = k + 1 K + 2 = 2N + 1, ω 2s ω k d k 2 Cε 2N+1. k >K M For the near-resonant modes we consider the rescaling 44, so that d k = ε [[k]] ω s k f k. We have f 2 1 Cε by 52 and 45, so that ω 2s,k R ε,h ω k d k 2 =,k R ε,h ω C sup,k R ε,h ω 2s 1 ω 2s 1 k ε2[[k]] ω 2 f k 2 2s 1 ε 2[[k]]+1 ω 2s 1 k. 17

18 The non-resonance condition 25 is formulated such that the supremum is bounded by C 2 0 ε2n+1, and hence ω 2s,k R ε,h 6.9 Defect in the non-resonant modes ω k d k 2 C ε 2N res-bound We now assume that k K and that,k satisfies the non-resonance condition 23, so that in the scaled variables c k of Section 6.4 the defect satisfies ω k d k = ε[[k]] 1 h 2 ψhω L k ck + fk c. Written in terms of the components a and b of 46 we have ω d ± ω k d k sinc hω ȧ± = ε ±2iεω ψhω = ε [[k]] 4 s +k s k h 2 ψhω + Aa ± ± + f a + b b k + Bb k + f k a + b. It should be noted that the functions in this defect are actually the 4N-th iterates a 4N and b 4N of the iteration in Section 6.3. Expressing f ± a+b and f k a+b in terms of Fa,b and Ga,b and inserting F and G from 47 into this defect, relates it to the increment of the iteration in the following way: ω d ± ω k d k = 2ω α ± = β k [ȧ± ] 4N [ȧ± [b k ] 4N [ b k ] 4N+1 ] 4N+1, α ± := ±iε 2 sinc hω ψhω, β k := ε[[k]] 4 s +k s k h 2 ψhω. Motivated by these relations we introduce new variables ã ± := α ± a ±, bk := β k bk. 58 var-xy Collecting these variables into vectors and using the transformed functions F ± ã, b := α ± F ± α 1 ã, β 1 b ± = ε f α 1 ã + β 1 b G k ã, b := βk ψhω ω + k ω Gk α 1 ã, β 1 b = ε [[k]] f k α 1 ã + β 1 b the iteration becomes ã n+1 = Ω 1 Fã, b Aã n b n+1 = Gã, b B b n ã n+1 0 = αv + P b n 0 + Qã n εh + Q b n εh. 59 xy-iter 18

19 In the iteration for the initial values we abbreviate P = αpβ 1, Q = αqβ 1, which are bounded by P b0 s C ε 1/2 b0 s Qãεh s C ε 1/2 sup ãτ s εh<τ<εh Q bεh s C ε 3/2 sup Ω 1 bτ s. εh<τ<εh In an H s neighbourhood of 0 where the bounds 51 hold, the partial derivatives of F with respect to ã and b and those of G with respect to b are bounded by Oε 1/2, whereas the derivatives of G with respect to ã is only O1. This is the same situation as we had for the exact solution in [6]. As in that paper one proves Ω ã 4N+1 τ ã 4N τ s C ε N+2 b 4N+1 τ b 4N τ s C ε N+2 ã 4N+1 0 ã 4N 0 s C ε N+2. These estimates yield the desired bound of the defect in the non-resonant modes,k R ε,h. Combined with the corresponding estimates of Subsection 6.8 we obtain 1/2 ω k d k τ s 2 Cε N+1 for τ d1 k K Consequently, the defect δt see Subsection 6.7 satisfies Ω 1 δt s+1 = δt s Cε N+1 for t ε defect-bound For the defect in the initial conditions 36 and 37 we obtain q0 q 0 s+1 + p0 p 0 s Cε N+1. For the alternative scaling ĉ k = ωs k z k, we obtain k K ω s k d k τ 2 1 1/2 Cε N+1 for τ ds 6.10 Remainder term of the modulated Fourier expansion We write the method 19 in the form q n+1 coshω sinhω q n Ω 1 p n+1 = sinhω coshω Ω 1 p n + h 2 Ψ 1 sinhωf n coshωf n + f n+1 where f n = Ω 1 fφq n, and we notice that Ψ 1 is a matrix, bounded independently of h and the dimension M. The differences q n := qt n q n and p n := pt n p n, where t n := nh, satisfy the same relation with f n replaced by Ω 1 fφ qt n fφq n +δt n. Using the Lipschitz bound cf. Section 4.2 in [13] on the relation between fq and gu of 1 Ω 1 fq 1 fq 2 s+1 = fq 1 fq 2 s Cε q 1 q 2 s Cε q 1 q 2 s+1 19

20 for q 1, q 2 H s satisfying q i s Mε, and the estimate 61 for the defect yields q n+1 Ω 1 p s+1 n+1 q n Ω 1 p s+1 n + h Cε q n s+1 + Cε q n+1 s+1 + Cε N+1. 2 Solving this inequality gives the estimate q n s+1 + Ω 1 p n s+1 C1 + t n ε N+1 for t n ε 1 and thus completes the proof of Theorem 6.1. sect:conservation 7 Conservation properties We now show that the system of equations determining the modulation functions has almost-invariants close to the actions, the momentum and the total energy along numerical solutions given by the full discretization The proof takes up arguments of [6] for the conservation of actions, of [13] for the conservation of momentum and aspects of the space discretization, and of [14, Ch. XIII] for the conservation of energy and for the aspects arising from the time discretization. 7.1 The extended potential The defect formula 55 can be rewritten as 1 h 2 ψhω L k zk + k UΦz = dk, 63 y-def-u where k Uy is the partial derivative with respect to y k potential see [13] of the extended Uy = U l y = N l= N N m=2 U l y 64 U U m+1 0 m + 1! k 1 + +k m+1 = m+1=2ml y k y km+1 m+1, where again k i 2N and i M, and Uu is the potential in Invariance under group actions The existence of almost-invariants for the system 63 turns out to be a consequence, in the spirit of Noether s theorem, of the invariance of the extended potential under continuous group actions: for an arbitrary real sequence µ = µ l l 0 and for θ R, let S µ θy = e ik µθ y k M, k K, Tθy = e iθ y k. 65 U-inv M, k K 20

21 Since the sum in the definition of U is over k k m+1 = 0 and that in U 0 over m+1 = 0, we have US µ θy = Uy, U 0 Tθy = U 0 y for θ R. Differentiating these relations with respect to θ yields 0 = d US µ θy = ik µy k dθ k Uy θ=0 k K M 0 = d U 0 Tθy = i y k dθ k U 0 y. θ=0 k K M 66 U-inv-diff 7.3 Almost-invariants of the modulation system We now multiply 63 once with ik µφhω z k and once with iφhω z k, and sum over and k with M and k K. Thanks to 66, we obtain k K M k K M φhω ik µ h 2 ψhω z k L k z k 67 Jmueq = ik µφhω z k k K M dk, φhω i h 2 ψhω z k L k z k 68 Meq = iφhω z k d k k U lφz. l =0 k K M By the expansion 35 of the operator L k, only expressions of the following type appear for zτ = z k τ and zτ = z k τ on the left-hand side of the above equations: Rezz 2l+1 = Re d dτ Im zz 2l+2 = Im d dτ zz 2l... ± z l 1 z l zl z l zz 2l+1 żz 2l +... ± z l z l+1. Therefore, the left-hand sides can be written as total derivatives of functions εj µ [z]τ and εk[z]τ which depend on zτ and its derivatives ε l z l τ for l = 1,...,N 1. In this way, 67 and 68 become ε d dτ J µ[z] = ε d dτ K[z] = k K M k K M 69 zbzp ik µφhω z k dk 70 Jmuder iφhω z k d k k U lφz. 71 Mder l =0 In the following we consider the special case of µ = l = 0,...,0, 1, 0,... with the only entry at the lth position and write J l [z] = J l [z]. From the smallness of the right-hand sides in 70 and 71 we infer the following. 21

22 thm:almost-inv Theorem 7.1 Under the conditions of Theorem 6.1 we have, for τ 1, M l=0 ω 2s+1 l d dτ J l[z]τ C ε N+1, d dτ K[z]τ C ε N+1 + ε 2 M s+1. Proof. The result is obtained from 70 and 71 with the arguments of [6, 13] as follows. With the bounds 52 and 62, the estimate for the functions J l [z] follows with the proof of Theorem 3 in [6]. With the bound 33 and with the bounds z 1 Cε and d 0 Cε N+1, which follow from 33 and 60, the estimate for K[z] is obtained as in Theorem 5.2 of [13]. A further almost-invariant is obtained by multiplying 63 with the expression φhω ik ωz k + εż k, summing over and k, and using 66: k K M φhω h 2 ψhω + ε d dτ UΦz = ik ωz k + εż k Lk z k 72 Heq k K M In addition to the identities 69 we also use Re żz 2l = Re d dt φhω ik ωz k + εż k żz 2l 1... z l 1 z l+1 ± 1 2 zl z l Im żz 2l+1 = Im d dtżz 2l zz 2l z l z l+1. d k. Therefore, the left-hand side of 72 can be written as the total derivative of a function εh[z]τ, so that 72 becomes ε d dτ H[z] = k K M φhω ik ωz k + εż k d k. 73 Hder As in Theorem 7.1, the Cauchy-Schwarz inequality and the estimates for z k and d k then yield the following estimate. thm:almost-invh Theorem 7.2 Under the conditions of Theorem 6.1 we have, for τ 1, d dτ H[z]τ C ε N Relationship with actions, momentum, and energy We now show that the almost-invariant J l of the modulated Fourier expansion is close to the corresponding harmonic action 13 of the numerical solution, J l = I l + I l = 2I l for 0 < l < M, J 0 = I 0, J M = I M, and that H and K are close to the Hamiltonian H M and the momentum K of 12 and 13, respectively. 22

23 thm:action Theorem 7.3 Under the conditions of Theorem 5.1, along the numerical solution q n, p n of 19 and the associated modulation sequence zεt, it holds that H[z]εt n = H M q n, p n + Oε 3 K[z]εt n = K q n, p n + Oε 3 + Oε 2 M s J l [z]εt n = J l q n, p n + γ l t n ε 3 with M l=0 ω2s+1 l γ l t n C for t n ε 1. All appearing constants are independent of ε, M, h, and n. Proof. With the identities 69 we obtain from 68 that K[z] = k K M φhω ψhω k ωsinc hk ω z k 2 +2εc 2k Im z k żk Separating the terms with k = ± and using the symplecticity condition 20, and applying the bounds 52 and 53 to the remaining terms, we find K[z] = ω z 2 z 2 + Oε 3. M In terms of the Fourier coefficients of the modulated Fourier expansion q t = k K zk εteik ωt, we have at t = t n K[z] = ω q + iω 1 p 2 q iω 1 p 2 + Oε 3 4 M = K q, p + Oε 3 + Oε 2 M s = Kq n, p n + Oε 3 + Oε 2 M s, where we have used 32. The Oε 2 M s terms come from the boundary terms in the sum. The last equality is a consequence of the remainder bound of Theorem 6.1. Similarly, we obtain from 72 that H[z] = k K M φhω k ω ψhω which yields, using in addition UΦz = Oε 3, M k ωsinc hk ω z k UΦz, H[z] = ω 2 z 2 + z 2 + Oε 3, and shows that H[z] = H M q n, p n + Oε 3. The result for J l is obtained in the same way, using in addition Lemma 3 of [6] to estimate the remainder terms. With an identical argument to that of [6, Section 4.5], Theorems 7.1 and 7.2 together with the estimates of Theorem 6.1 and the Lipschitz continuity 54 yield the statement of Theorem 5.1 by patching together many intervals of length ε 1. 23

24 sect:leapfrog 8 The Störmer Verlet/leapfrog discretization The leapfrog discretization of 11 reads, in the two-step formulation, q n+1 2 q n + q n 1 = h 2 Ω 2 q n + fq n, 74 verletu with the velocity approximation p n given by 2h p n = q n+1 q n verletv The starting value is chosen as q 1 = q 0 +hp 0 + h2 2 fq0. Conservation properties of this method will be obtained by reinterpreting it as a trigonometric method 17 with modified frequencies ω satisfying h2 ω 2 = cosh ω, that is, 1 sin h ω = 1 hω. 76 omegatilde 2 2 This is possible as long as hω 2. thm:conserve-sv Theorem 8.1 Under the stepsize restriction hω M c < 2, under the nonresonance conditions 25 and 27 for the modified frequencies ω of 76, and under the assumption 15 of small initial data with s σ + 1 for q 0, p 0 = q0, p0, the near-conservation estimates M l=0 H M q n, p n H M q 0, p 0 ε 2 Cε + h 2 Kq n, p n Kq 0, p 0 ε 2 C ε + h 2 + M s + εtm s+1 ω 2s 1 l I l q n, p n I l q 0, p 0 ε 2 Cε + h 2 for energy, momentum and actions hold for long times 0 t = nh ε N+1 with a constant C which depends on s, N, C 0, and c, but is independent of ε, M, h, and t. Proof. Denoting by Ω the diagonal matrix with entries ω, we introduce the transformed variables q n = sinch Ωq n, p n = p n, which are solutions to the symplectic trigonometric method with ψ = sinc and φ = 1. Under the stepsize restriction hω M c < 2 the non-resonance condition 26 is trivially satisfied for ω, and we have ω ω Cω, where C depends only on c. Hence, the assumption 15 of small initial data is satisfied with the same exponent s for the weighted norms defined with ω or ω. We can therefore apply Theorem 5.1 in the transformed variables q n, p n. With the estimate sinc h ω h2 ω 2, the result stated for the original variables q n, p n then follows. 24

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