Uniformly accurate averaging numerical schemes for oscillatory evolution equations

Transcription

1 Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University of Rennes) and G. Vilmart (University of Geneva) Rennes, March 29, 2018

2 Highly-oscillatory problems with periodic time-dependence Highly-oscillatory problems with periodic time-dependence We consider a highly oscillatory problem in a (functional) Banach space X: (P ε) d dt uε (t) = f t/ε (u ε (t)), u ε (t) X, u ε (0) = u 0 X, where ε is a possibly small parameter (scales as the inverse of a frequency). (θ, u) f θ (u) is given, smooth and P -periodic with respect to θ. Assumption: There exist T > 0 and a bounded open subset K X s.t., for all ε ]0, 1], (P ε) admits a unique solution u C 1 ([0, T ], K). Numerical difficulties: Standard schemes lead to u ε u ε, t C ( t)p ε q, q > 0, forcing t < ε q/p and thus formidable costs for small values of ε. Partial remedy: Averaging methods lead to u ε u ε, t C(( t) p + ε q ). Aim: Construct uniformly accurate methods, i.e. u ε u ε, t C( t) p.

3 Outline of the talk 1 Introduction 2 Averaging methods 3 Micro-macro numerical method 4 Numerical results

4 Examples (i) Our examples are of the form d dt yε = 1 ε Ayε + f(y ε ) where θ e θa is periodic with respect to θ. To filter out the highly oscillatory dynamics, we introduce the unknown u ε (t) = e t ε A y ε (t) which satisfies Here we have d ( ) dt uε = e t A ε f e ε t A u ε (t) = f t (u ε (t)) ε f θ (u) = e θa f ( ) e θa u ε (t)

5 Examples (ii) The nonlinear Klein-Gordon equation in the nonrelativistic regime 1 ε ttu ε u ε + 1 ε uε + f(u ε ) = 0, x R d, t > 0, Schrödinger equation and Gross-Pitaevskii equation i tψ ε = 1 ε ψε + α ψ ε 2 ψ ε with periodic boundary conditions i tψ ε = 1 ε ψε + ω ε x 2 ψ ε + α ψ ε 2 ψ ε on R d Vlasov-Poisson in a strong uniform magnetic field tf ε + v xf ε + E vf ε + v B ε vf ε = 0 1 Bao-Dong 2012

6 Averaging results (i) Averaging methods assert that (for sufficiently small ε) there exist 1 a P -periodic change of variables (τ, u) T X Φ ε τ (u) X 2 a smooth, autonomous vector field and its flow-map such that t [0, T ], u X F ε (u) X (t, u) Ψ ε t (u) X d dt Ψε t (u) = F ε (Ψ ε t (u)) u ε (t) Φ ε t Ψ ε t (Φ ε ε 0) 1 (u 0) Ce C ε. X high-oscillations slow drift perturbation ref.: Bogoliubov-Mitropolsky 1930, Perko 1968,... and their adaptation to PDEs: JFOCM 15, Castella-C.-Méhats-Murua

7 Averaging results (ii) Various possibilities exist for the change of variables (it is not unique). Two choices have become prominent: in classical averaging (such as used in KAM theory), one takes Φ ε ( ) := 1 Φ ε τ ( )dτ = id; T in the so-called stroboscopic averaging, one imposes T Φ ε 0( ) = id. Second choice entails two advantages: it preserves geometric properties and it is amenable to numerical methods.

8 Averaging (iii): equations for averaging By differentiating with respect to time t u ε (t) = Φ ε t/ε Ψ ε t (Φ ε 0) 1 (u 0) = Φ ε t/ε Ψ ε t (ǔ 0), ǔ 0 = (Φ ε 0) 1 (u 0), and remembering that we obtain formally d dt Ψε t (u) = F ε (Ψ ε t (u)) 1 ε θφ ε t/ε(ψ ε t (ǔ 0)) + uφ ε t/ε(ψ ε t (ǔ 0))F ε (Ψ ε t (ǔ 0)) = f t/ε (Φ ε t/ε(ψ ε t (ǔ 0)) and then by taking ǔ 0 = Ψ ε t(u) θ Φ ε θ(u) + ε uφ ε θ(u)f ε (u) = εf θ Φ ε θ(u). Taking the average in θ of this equation yields F ε = uφ ε 1 f Φ ε This is a closed system of equations on Φ ε θ and F ε... which has no exact solution in general (Neistadt). However, one can use it to construct two sequences (Φ [n] θ ) n 0, (F [n] ) n 0 which approximate Φ ε θ and F ε

9 Averaging (iv): truncations Starting from Φ [0] = id we define for k = 0, 1,... F [k] = uφ [k] 1 f Φ [k], Φ [k+1] θ = id X εg [k+1] +ε θ 0 ( f τ Φ [k] τ uφ [k] τ F [k]) dτ where G [k+1] is chosen to ensure that Φ ε 0 = id X or Φ ε = id X. Then F [k] and satisfy previous closed system with (zero average) defect Φ [k+1] θ δ [k] (u) = 1 ε θφ [k] (u) + uφ[k] (u)f [k] (u) f θ Φ [k] (u). θ θ θ Theorem (Castella, C., Murua and Méhats, JFOCM 15) Under appropriate regularity assumptions, there exists ε 0 > 0 such that the maps Φ [k] and F [k] are well-defined for (k + 1)ε ε 0, Φ [k] is continuously differentiable w.r.t. θ and there exists M > 0 such that sup δ [k] θ (2(k M + 1) ε ) k. θ T ε 0 By choosing k as a function of ε one obtains the Ce C ε error term of averaging.

10 Averaging (v): stroboscopic versus standard averagings From the stroboscopic decomposition u ε (t) = Φ ε t/ε Ψ ε t (u 0) with Φ ε 0 = id X, we may write (once again formally at this stage) u ε (t) = Φ ε t/ε Φ ε 1 Φ ε Ψ ε t Φ ε 1 Φ ε (u 0). }{{}}{{}}{{} Φ ε Ψ ε t/ε t ( Φ ε 0 ) 1 It can indeed be checked, on the one hand, that and and on the other hand, that d dt Ψ ε t (u 0) = Φ ε = Φ ε Φ ε 1 = Φ ε Φ ε 1 = id X Φ ε 0 = Φ ε 0 Φ ε 1 = id X Φ ε 1 = Φ ε 1, ( ( u Φ ε F ) Φ ε 1) Ψ ε t (u 0) = f Φ ε Ψ ε t (u 0) = F ε Ψ ε t (u 0). The maps Φ ε, Ψ ε and F ε correspond to standard averaging and we have δ ε θ = δ ε θ Φ ε 1. Conversely, we may relate Φ ε to Φ ε through the formula Φ ε θ = Φ ε θ ( Φ ε 0) 1.

11 Averaging (vi): first terms for stroboscopic averaging Step 0: Φ [0] θ (u) = u, F [0] (u) = f (u) Step 1: Φ [1] θ Previous theorem shows that θ (u) = u + ε (f τ (u) f ) dθ 0 F [1] (u) = uφ [1] 1 f Φ [1] = f ε τ [f τ (u), f σ(u)]dσdτ + O(ε 2 ) 2P u ε (t) = Φ [k] t ε T 0 Ψ [k] t (u 0) + O(ε k+1 ). In the limit ε 0, we have u ε (t) = Ψ 0 t (u 0) + O(ε) with (P 0) : d dt Ψ0 (t) = f ( Ψ 0 (t) ), Ψ 0 (0) = u 0 X.

12 Averaging (vi): an assessment Standard use of averaging methods consists in the simulation of d dt Ψε t = F [k] (Ψ ε t ) (possibly) complemented with the computation of Φ [k] θ. Pros of averaging models are non-stiff and do not suffer from severe constraints on the time step when ε is small preserve part or all geometric structures for stroboscopic averaging Cons for values of ε away from 0 one needs to include many terms in the expansion leading to important costs methods based on the (numerical or not) computation of the averaged vector field F ε lead to an incompressible error term owing to the truncation of the series

13 Uniformly Accurate schemes Main drawback of an Asymptotic-Preserving schemes intermediate regimes may suffer from order reduction. Our aim is ow to construct here uniformly accuracy (UA) schemes with errors u ε, t u ε sup C( t) p, ε [0,1] u ε where p is the order of the numerical method and C does not depend on ε. Computational cost must not degrade when ε 0 and a certain number of corrective terms in ε should be captured by the method (not only the limit P 0).

14 New approach: micro-macro method (i) If we insist on solving an autonomous equation ( ) (u) = F [k] Ψ [k] (u) Ψ [k] t t, Ψ [k] 0 (u) = u, then we retain the information contained in the remainder G [k] t (u 0) = u ε (t) Φ [k] t/ε Ψ[k] t (Φ [k] 0 ) 1 (u 0), which obeys the following differential equation d dt G[k] t ( (u 0) = u ε 1 (t) ε θφ [k] = f t/ε ( Φ [k] t/ε Ψ[k] t with ǔ 0 = (Φ [k] 0 ) 1 (u 0). t/ε + uφ[k] t/ε F [k]) (Ψ [k] t (ǔ 0)) ) ( ) (ǔ 0) + G [k] t (u 0) f t/ε Φ [k] t/ε Ψ[k] t (ǔ 0) δ [k] t/ε (Ψ[k] t (ǔ 0))

15 New approach: micro-macro method (ii) The remainder w ε = G [k] t ẇ ε (t) = f θ ( Φ [k] θ θ(t) = t/ε, ǔ 0 = (Φ [k] 0 ) 1 (u 0), w ε (0) = 0. (u 0) solves the micro equation ) ( f θ Ψ [k] t (ǔ 0) + w ε (t) Φ [k] θ Ψ [k] t (ǔ 0) ) δ [k] θ (Ψ[k] t (ǔ 0)), One can rewrite the micro-macro system as follows: v ε = F [k] (v ε ), v ε (0) = (Φ [k] 0 ) 1 (u 0) ( w ε = f t/ε Φ [k] t/ε (vε ) + w ε) ( ) f t/ε Φ [k] t/ε (vε ) δ [k] t/ε (vε ), w ε (0) = 0.

16 New approach: Pullback method If we insist on satisfying the equation (P ) : u ε (t) = Φ [k] t/ε Ψ[k] t (Φ [k] 0 ) 1 (u 0), then the autonomous equation can not hold and should instead be amended to Ψ [k] t (ǔ 0) = F [k] Ψ [k] (ǔ 0) + R [k] By differentiating (P ) we have t t/ε (Ψ[k] t (ǔ 0)), Ψ [k] (ǔ0) = ǔ = u ε (t) 1 ε θφ [k] t/ε (Ψ[k] t (ǔ 0)) uφ [k] t/ε (Ψ[k] t (ǔ 0)) Ψ [k] t (ǔ 0) = uφ [k] t/ε (Ψ[k] t (ǔ 0)) (F [k] (Ψ [k] (ǔ 0)) t [k] Ψ t (ǔ 0)) 1 ε δ[k] t/ε (Ψ[k] t (ǔ 0)), and we get R [k] θ ( = uφ [k] θ ) 1 δ [k] This leads to the following pulled back equation ( ) 1 ( v = uφ [k] t/ε f t/ε Φ [k] t/ε (v) 1 ) ε θφ [k] t/ε (v), v(0) = (Φ [k] 0 ) 1 (u 0), (1) from which the solution can be recovered as u ε (t) = Φ [k] t/ε (v(t)). θ.

17 The four methods Both the micro-macro and the pullback formulations are useful for the design of uniformly accurate numerical schemes. They can be considered for stroboscopic and standard averagings, yielding four different approaches. Pullback method Micro-macro method Stroboscopic averaging Standard averaging Φ [k] 0 = id X, G [k] 0 Φ [k] = id X, G [k] 0 Φ [k] 0 = id X, R [k] 0 Φ [k] = id X, R [k] 0 Table: Designing choices for uniformly accurate averaging. The main argument underlying our approach is now the following: Theorem (C., Lemou, Méhats and Vilmart, 2017) If f is C p w.r.t. θ, Φ [k] and δ [k] are resp. C p+1 and C p w.r.t. θ for (k + 1)ε ε 0. Moreover, there exists M > 0 such that 0 ν p, sup θ ν δ [k] θ (M(k M + 1) ε ) k. θ T ε 0

18 A word on the boundedness of time-derivatives Now, it is apparent that the micro-macro system is of the form where: v ε = F [k] (v ε ), v ε (0) = u 0 ẇ ε = A [k] θ (vε, w ε )w ε + b θ (v ε ), w ε (0) = 0 θ = t/ε 1 A and b are smooth w.r.t. v, w if f is itself sufficiently smooth w.r.t. u; 2 A and b are p-times differentiable w.r.t. θ if f is; 3 the k th -derivative of b w.r.t. θ up to k = p are bounded by ε k. Fundamental consequence (from Gronwall lemma): uniform boundedness If f is of class C p w.r.t. θ with p k, then ε ]0, 1], 0 s k + 1, t [0, T ], ds dt s wε Cεk+1 s

19 Advantages of this method 1 It is clear that if Ψ ε and w ε satisfy the micro/macro equations, then u ε (t) = Φ [k] t/ε (Ψε t (ǔ 0)) + w ε (t) satisfies the original equation d dt uε (t) = f t/ε (u ε (t)). In contrast with usual averaging, our micro/macro method is not an approximation and contains whole the information of the original problem. 2 The fact that w ε has bounded time-derivatives with respect to ε allows to use standard numerical methods for the micro/macro system with uniform accuracy with respect to ε.

20 Advantages of this method 1 It is clear that if Ψ ε and w ε satisfy the micro/macro equations, then u ε (t) = Φ [k] t/ε (Ψε t (ǔ 0)) + w ε (t) satisfies the original equation d dt uε (t) = f t/ε (u ε (t)). In contrast with usual averaging, our micro/macro method is not an approximation and contains whole the information of the original problem. 2 The fact that w ε has bounded time-derivatives with respect to ε allows to use standard numerical methods for the micro/macro system with uniform accuracy with respect to ε.

21 Uniformly accurate numerical schemes applied to the micro-macro system A standard numerical scheme applied to the equation is of conventional order p if d dt y(t) = gε (t/ε, y(t)) 0 < ε < ε 0, C ε, 0 < t < t 0, y n y(t n ) C ε t p. It is of uniform order p if C > 0, 0 < ε < ε 0, 0 < t < t 0, y n y(t n ) C t p. If y(t) and g ε have (p + 1) derivatives uniformly bounded w.r.t. ε, then a standard method of conventional order p is also of uniform order p. Take-away message: A p-th order standard scheme is of uniformly accurate of order p when applied to the micro-macro system provided Φ [p] and F [p] are used.

22 Integral numerical scheme In practice, it is better to use a scheme based on Duhamel formulation: t n+1/2 y n+1/2 = y n + y n+1 = y n + t n t n+1 t n g ε (t/ε, y n )dt g ε (t/ε, y n+1/2 )dt. Note that the function g being periodic with respect to its first variable, the integral can be computed numerically spectrally by using Fourier decompositions. For this problem, one only needs two derivatives of y and one derivative in time of g. Here, Φ [1] and F [1] are sufficient to build the micro/macro model.

23 The Hénon-Heiles model We first consider the Hénon-Heiles Hamiltonian model parametrized with ε: H(q 1, q 2, p 1, p 2) = p2 1 2ε + p q2 1 2ε + q q2 1q q3 2 Which is highly oscillatory for small ε is small. The associated filtered system, satisfied by the variable u ε (t) R 4 defined by ( ) ( ) u ε t t 1(t) = cos q 1(t) sin p 1(t), ε ε u ε 2(t) = q 2(t), ( ) ( ) u ε t t 3(t) = sin q 1(t) + cos p 1(t), ε ε u ε 4(t) = p 2(t). This variable u ε (t) satisfies the system with du ε dt (t) = f f 1(θ, u) = 2 sin θ (u 1 cos θ + u 3 sin θ) u 2 f 2(θ, u) = u 4 ( ) t ε, uε (t), (2) f 3(θ, u) = 2 cos θ (u 1 cos θ + u 3 sin θ) u 2 f 4(θ, u) = 2 (u 1 cos θ + u 3 sin θ) 2 + u 2 2 u 2.

24 Error versus t for ε = 2 k, k {0, 1, 2,, 9} Figure: Left: Micro-macro method. Right: Pullback method.

25 Hénon-Heiles, pullback method, long time error on the energy E Figure: Left: ε = 1. Right: ε = Blue: order 2, t = 0.2. Red: order 2, t = 0.1. Black: order 3, t = 0.2.

26 Introduction Averaging methods Micro-macro numerical method Numerical results Poincare cuts for He non-heiles model. Pullback method with midpoint rule Figure: Left: ε = 1, H = 1/12. Right: ε = 0.001, H = 1/

27 The nonlinear Klein-Gordon equation in the nonrelativistic regime (i) This system reads ε ttu ε u ε + 1 ε uε + f(u ε ) = 0, x R d, t > 0, with initial conditions given as u ε (0, x) = φ(x), tu ε (0, x) = 1 ε γ(x), x Rd. Let us write the equivalent first order system satisfied by the unknown v+ ε = u ε iε(1 ε ) 1/2 tu ε, v ε = u ε iε(1 ε ) 1/2 tu ε, Setting f(v +, v ) = (f( 12 (v+ + v )), f( 12 (v+ + v )) ), we obtain i tv ε = 1 ε (1 ε )1/2 v ε (1 ε ) 1/2 f(v ε ), v ε = (v ε +, v ε ).

28 The filtered form of the Klein-Gordon equation (ii) Let us introduce the filtered unknown ũ ε = e i ε t v ε. This quantity satisfies: i tũ ε + A ε ũ ε = (1 ε ) 1/2 e i ε t f (e ) i ε t ũε with A ε = 1 ε ( 1 ε 1 ). This self-adjoint operator is not singular as ε 0, indeed, for all ε > 0, we have in the sense of operators. 0 A ε 1 2

29 Numerical tests for the nonlinear Klein-Gordon system We test the following numerical methods: Our second order method: UA of order 2 A third order method constructed with Φ [2] and F [2] by extrapolation: UA of order 3 Note: the vector fields cannot be precomputed easily and the method contains an additional variable θ on which the averaged are computed. For instance 1 2π 2π 0 i(1 ε ) 1/2 e iθ f ( ) e iθ u dθ is computed by the rectangle formula (with spectral accuracy).

30 NKG model, error versus t for ε = 10 k, k {0, 1, 2,, 6} Figure: Left: Micro-macro method. Right: Pullback method.

31 NKG model, pullback method, long-time errors on the invariant Q and energy E 1 x x error on Q 0 1 error on E time time Figure: Blue: order 2 with t = Red: order 2 with t = Black: order 3 with t = 0.01.

32 Work under completion In the paper Oscillatory evolution problems with degenerate time-dependent frequency, C., Lemou, Méhats and Vilmart, to be submitted the following situation is envisaged: where U ε (t) = γ(t) ( ) ε AU ε (t) + f U ε (t) R d, U ε (0) = u 0 R d, t 0. ε lies in (0, 1] A is supposed to be diagonalizable and to have only eigenvalues in iz γ vanishes at some instant t 0 Under these circumstances: an asymptotic analysis can be carried out uniformly accurate numerical methods can be constructed This situation is not covered by the standard theory of averaging.

33 Thank you for your attention