Sympletic Methods for Long-Term Integration of the Solar System

Transcription

1 Sympletic Methods for Long-Term Integration of the Solar System A. Farrés J. Laskar M. Gastineau S. Blanes F. Casas J. Makazaga A. Murua ( ) Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris Instituto de Matemática Multidisciplinar, Universitat Politècnica de València Institut de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I Konputazio Zientziak eta A.A. saila, Informatika Fakultatea 22 Abril 2013 Seminari Informal de Matemàtiques de Barcelona (SIMBa)

2 Overview of the Talk 1 Why do we want long-term integrations of the Solar System? 2 The N-Body Problem (Toy model for the Planetary motion) 3 Symplectic Splitting Methods for Hamiltonian Systems A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

3 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

4 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

5 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

6 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

7 Planetary Solution La2004 : numerical, simplified, tuned to DE406 (6000 yr) INPOP : numerical, complete, adjusted to observations. 1 Myr : 6 months of CPU. La2010 : numerical, less simplified, tuned to INPOP (1 Myr ). 250Myr : 18 months of CPU. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

8 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

9 Numerical Precision La2010a is fine for 60 Myr But 18 months of CPU for 250 Myr! (Laskar et al, 2010) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

10 For further information A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

11 The Challenge 1 The NUMERICAL PRECISION of the solution. We want to be sure that the precision is not a limiting factor. 2 The SPEED of the algorithm. As La2010a took nearly 18 months to complete. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

12 The N - Body Problem

13 The N-Body Problem We consider that we have n + 1 particles (n planets + the Sun) interacting between each other due to their mutual gravitational attraction. We consider: u 0, u 1,..., u n and u 0, u 1,..., u n the position and velocities of the n + 1 bodies with respect to the centre of mass. ũ i = m i u i the conjugated momenta. The equations of motion are Hamiltonian: H = 1 2 n i=0 ũ i 2 m i G 0 i<j n m i m j u i u j. (1) Notice that the Hamiltonian is naturally split as H = T (p) + U(q). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

14 The N-Body Problem (Planetary Case) In an appropriate set of coordinates: H = H A (p, q) + εh B (q) H = H A (a) + εh B (a, λ, e, ω, i, Ω) Where H A corresponds to the Keplerian motion and H B to the Planetary interactions. Change of variables: (p, q) (a, λ, e, ω, i, Ω) (Wisdom & Holman, 1991 Kinoshita, Yoshida, Nakai, 1991) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

15 Jacobi Coordinates We consider the position of each planet (P i ) w.r.t. the centre of mass of the previous planets (P 0,..., P i 1 ). v 0 = (m 0u m nu n)/η n v i = u i ( i 1 j=0 m ju j )/η i 1 }, ṽ 0 = ũ ũ n ṽ i = (η i 1 ũ i m i ( i 1 j=0 u j))/η i }. where η i = i j=0 m j. In this set of coordinates the Hamiltonian is naturally split into two part: H J = H Kep + H pert : n ( 1 η i ṽ i 2 H J = G m ) iη i 1 n ( ) + G ηi 1 m i 2 η i 1 m i v i v i m0 m i m j r i ij i=1 where i,j = u i u j. i=2 0<i<j n, A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

16 Heliocentric Coordinates We consider relative position of each planet (P i ) with respect to the Sun (P 0 ). } },, r 0 = u 0 r i = u i u 0 r 0 = ũ ũ n r i = ũ i In this set of coordinates the Hamiltonian is naturally split into two part: H H = H Kep + H pert : H H = n i=1 ( 1 2 r i 2 where i,j = r i r j. [ m0 + m i m 0m i ] G m0m ) i + ( ri r j G m ) im j, r i m 0 ij 0<i<j n A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

17 Jacobi Vs Heliocentric coordinates In both cases we have H = H Kep + H pert. But: - H H = H A (p, q) + ε(h B (q) + H C (p)), - H J = H A (p, q) + εh B (q), where H A, H B and H C are integrable on their own. Remarks: the size of the perturbation in Jacobi coordinates is smaller that the size of the perturbation in Heliocentric coordinates, giving a better approximation of the real dynamics. the expressions in Heliocentric coordinates are easier to handle, and do not require a specific order on the planets. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

18 Jacobi Vs Heliocentric (size of perturbation) np,case Heliocentric Pert. Jacobi Pert. 2, MV E E-011 2, JS E E-007 4, MM E E-010 4, JN E E-007 8, MN E E-007 8, VP E E-007 9, All E E-007 Table: Size of the perturbation in Heliocentric Vs Jacobi coordinates for different type of planetary configurations. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

19 Jacobi Vs Heliocentric coordinates i, j Heliocentric Pert. Jacobi Pert. 1, E E-011 2, E E-010 3, E E-011 4, E E-010 5, E E-007 6, E E-008 7, E E-009 8, E E-013 Table: Size of the perturbation in Heliocentric Vs Jacobi coordinates for the consecutive pair of planets. Here, 1 = Mercury, 2 = Venus, 3 = Earth-Moon Barycentre, 4 = Mars, 5 = Jupiter, 6 = Saturn, 7 = Uranus, 8 = Neptune, 9 = Pluto. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

20 Symplectic Splitting Methods for Hamiltonian Systems

21 Splitting Methods for Hamiltonian Systems Let H(q, p) be a Hamiltonian, where (q, p) are a set of canonical coordinates. dz dt = {H, z} = L Hz, (2) where z = (q, p) and {, } is the Poisson Bracket ({F, G} = F qg p F pg q). The formal solution of Eq. (2) at time t = τ that starts at time t = τ 0 is given by, z(τ) = exp(τl H )z(τ 0). (3) The main idea is to build approximations for exp(τl H ) that preserve the symplectic character. We focus on the special case H = H A + εh B, where H A and H B are integrable on its own. This is the case of the N-body planetary system, where the system can be expressed as a Keplerian motion plus a small perturbation due to their mutual interaction. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

22 Splitting Methods for Hamiltonian Systems The formal solution of Eq. (2) at time t = τ that starts at time t = τ 0 is given by, where A L HA, B L HB. z(τ) = exp(τl H )z(τ 0) = exp[τ(a + εb)]z(τ 0). (4) We recall that H A and H B are integrable, hence we can compute exp(τa) and exp(τ B) explicitly. We will construct symplectic integrators, S n (τ), that approximate exp[τ(a + εb)] by an appropriate composition of exp(τ A) and exp(τ εb): S n (τ) = n exp(a i τa) exp(b i τεb) i=1 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

23 Splitting Methods for Hamiltonian Systems Using the Baker-Campbell-Hausdorff (BHC) formula for the product of two exponential of non-commuting operators X and Y : with exp X exp Y = exp Z, Z = X + Y [X, Y ] ([X, [X, Y ]] [Y, [Y, X ]]) + 1 [X, [Y, [Y, X ]]] +..., 24 and [X, Y ] := XY YX. This ensures us that is we have an nth order integrating scheme: k exp(a i τa) exp(b i τb) = exp(τd H ). i=1 Then H = H + τ n H n + o(τ n ) and the error in energy is of order τ n. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

24 Two simple examples S 1 (τ) = exp(τa) exp(τb), K = A + B + τ 2 τ [A, B] + ([A, [A, B]] + [B, [B, A]]) S 2 (τ) = exp(τ/2a) exp(τb) exp(τ/2a), K = A + B + τ 2 ([A, [A, B]] + [B, [B, A]]) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

25 Many Authors like Ruth(1983), Neri (1987) and Yoshida(1990) among others have found appropriate set coefficientscients a i, b i in order to have a High Order symplectic integrator (4th, 6th, 8th,...). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

26 Splitting Methods for Hamiltonian Systems Let us call S n (τ) = exp(τk). Where, n S n (τ) = exp(a i τa) exp(b i τεb) = exp(τk), (5) i=1 The BCH theorem ensures us that K L({A, B}), the Lie algebra generated by A and B, and it can be expanded as a double asymptotic series in τ and ε: τk = τp 1,0A + ετp 1,1B + ετ 2 p 2,1[A, B] + ετ 3 p 3,1[A, [A, B]] + ε 2 τ 3 p 3,2[B, [B, A]] + ετ 4 p 4,1[A, [A, [A, B]]] + ε 2 τ 4 p 4,2[A, [B, [B, A]]] + ε 3 τ 4 p 4,3[B, [B, [B, A]] +..., where p i,j are polynomials in a i and b i. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

27 Splitting Methods for Hamiltonian Systems We will say that a method S n (τ) has order p if K = A + εb + o(τ p ). Hence, the coefficients a i, b i must satisfy: p 1,0 = 1, p 1,1 = 1, p i,j = 0, for i = 2,..., p. Remark: It is easy to check that, p 0,1 = a 1 + a a n = 1, p 1,1 = b 1 + b b n = 1. If S n (τ) = S n ( τ) then all the terms of order τ 2k+1 are cancelled out. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

28 Splitting Methods for Hamiltonian Systems S n(τ) = n exp(a i τa) exp(b i ετb) = exp(τk), i=1 In general ε τ (or at least ε τ), so we are more interested in killing the error terms with small powers of ε. We will find the coefficient a i, b i such that: τk τ(a + εb) = O(ετ s ε 2 τ s ε 3 τ s ε m τ sm+1 ). (6) Definition We will say that the method S n (τ) has n stages if it requires n evaluations of exp(τa) and exp(τb) per step-size. Definition We will say that the method S n (τ) has order (s 1, s 2, s 3,...) if it satisfies Eq. (6). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

29 SABA n or McLachlan (2n,2) methods McLachlan, 1995; Laskar & Robutel, 2001, considered symmetric schemes that only killed the terms of order τ k ε for k = 1,..., 2n. S m (τ) = exp(a 1 τa) exp(b 1 τb)... exp(b 1 τb) exp(a 1 τa). The main advantages are that: We only need n stages to have a method of order (2n, 2). We can guarantee that for all n the coefficients a i, b i will always be positive. - McLachlan, 1995: Composition methods in the presence of small parameters, BIT 35(2), pp Laskar & Robutel, 2001: High order symplectic integrators for perturbed Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy 80(1), A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

30 SABA n or McLachlan (2n,2) methods McLachlan, 1995; Laskar & Robutel, 2001 id order stages a i b i SABA1 or ABA22 (2, 2) 1 a 1 = 1/2 b 1 = 1 a SABA2 or ABA42 (4, 2) 2 1 = 1/2 3/6 a 2 = b 1 = 1/2 3/3 a SABA3 or ABA62 (6, 2) 3 1 = 1/2 15/10 a 2 = b 1 = 5/18 15/10 b 2 = 4/9 a 1 = 1/ ( 30/70 a 2 = SABA4 or ABA82 (8, 2) ) b 1 = 1/4 30/72 30 /70 b 2 = 1/4 + 30/72 a 3 = /35 SBAB1 or BAB22 (2, 2) 1 a 1 = 1 b 1 = 1/2 b SBAB2 or BAB42 (4, 2) 2 a 1 = 1/2 1 = 1/6 b 2 = 2/3 a SBAB3 or BAB62 (6, 2) 3 1 = 1/2 5/10 a 2 = b 1 = 1/12 5/5 b 2 = 5/12 SBAB4 or BAB82 (8, 2) 4 a 1 = 1/2 3/7/2 a 2 = 3/7/2 b 1 = 1/20 b 2 = 49/180 b 3 = 16/45 Table: Table of coefficients for the ABA, BAB methods of order (2s, 2) for s = 1,..., 4. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

31 SABA n or McLachlan (2n,2) methods SABA2 SABA3 SABA4 SBAB2 SBAB3 SBAB4 Mercury - Venus (Jacobi Coord) SABA2 SABA3 SABA4 SBAB2 SBAB3 SBAB4 Jupiter - Saturn (Jacobi Coord) Figure: Comparison of the performance of the SABA n and SBAB n schemes for the couples Mercury - Venus (left) and the Jupiter-Saturn (right) (Jacobi Coordinates) In log scale maximum error energy Vs. cost (τ/n). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

32 SABA n or McLachlan (2n,2) methods As we have seen in the figures above, the main limiting factor of these methods are the terms of order τε 2, which become relevant when τ is small. We recall that in the methods described above we have: K = (A + εb) + ετ 2n p 2n,1[A, [A, [A, B]]] + ε 2 τ 2 p 3,2[B, [B, A]] +..., There are in the literature several options to kill the terms of order τ 2 ε 2 {{A, B}, B}. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

33 Symplectic Integrator (killing the terms of higher order) Let S 0 (τ) be any of the given symmetric symplectic schemes previously described: S 0(τ) = exp(a 1τA) exp(b 1τB)... exp(b 1τB) exp(a 1τA) = exp(τk), where K = (A + εb) + ετ 2n p 2n,1[A, [A, [A, B]]] + ε 2 τ 2 p 3,2[B, [B, A]] In order to kill the terms of order ε 2 τ 2 we can: 1 Add a corrector term: exp( τ 3 ε 2 c/2l C )S 0 (τ) exp( τ 3 ε 2 c/2l C ). 2 Composition method: S m 0 (τ)s 0(cτ)S m 0 (τ), where c = (2m) 1/3. 3 Add extra stages: S(τ) = m i=1 exp(a iτa) exp(b i τb), with m > n. Hence, the reminder will be τ 2n ε + τ 4 ε 2, having methods of order (2n, 4). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

34 The corrector term L C This option was proposed by Laskar & Robutel, K = (A + εb) + ε 2 τ 2 p 3,2[B, [B, A]] + ετ 2n p 2n,1[A, [A, [A, B]]] +..., Notice that if A is quadratic in p and B depends only of q then [B, [B, A]] is integrable. We will consider SC n (τ) = exp( τ 3 ε 2 b/2l C )S n (τ) exp( τ 3 ε 2 b/2l C ), with C = {{A, B}, B}. order c ABAn c BABn 1 1/12 1/24 2 (2 3)/24 1/72 3 ( )/648 (13 5 5)/ ( )/64800 REMARK: This procedure only works in Jacobi coordinates. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

35 Composition method The idea behind this option was first discussed by Yoshida (1990). generalise He showed that if S(τ) is a symplectic methods of order 2k, then it is possible to find a new method of order 2k + 2 by taking where c must satisfy, c 2k = 0. We can generalise these as: where now, c = (2m) 1/(2k+1). S(τ)S(cτ)S(τ), S m (τ)s(cτ)s m (τ), With this simple composition methods we can transform any of the (2s, 2) methods described above to (2s, 4) method. REMARK: This procedure works for both set of coordinates. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

36 Adding an extra stage (McLachlan (2s,4)) McLachlan discussed the possibility of adding an extra stage to methods of order (2s, 2) in order to get rid of the ε 2 τ 2 terms:. n+1 S(τ) = exp(a i τa) exp(b i τb) i=1 id order stages a i b i ABA64 (6, 4) 4 BAB64 (6, 4) 4 ABA84 (8, 4) 5 BAB84 (8, 4) 5 a 1 = a 2 = a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = Notice that we no longer have positive values for the coefficients a i, b i. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

37 Jacobi Coordinates (first results) Mercury - Venus - Earth - Mars (Jacobi Coord) SABA4 (8,2) SABA4 m=2 (8,4) SABAC4 (8,4) McLa ABA (8,4) Jupiter - Saturn - Uranus - Neptune (Jacobi Coord) SABA4 (8,2) SABA4 m=2 (8,4) SABAC4 (8,4) McLa ABA (8,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

38 A couple of REMARKS!!

39 Remark 1: Splitting Methods in Heliocentric Coordinates We recall that in Heliocentric coordinates: H(p, q) = H A (p, q) + ε(h B (q) + H C (p)). We can use the same integrating schemes introduced above: n S(τ) = exp(a i τa) exp(b i τ(b + C)), i=1 We can use the approximation: exp(τ(b + C)) = exp(τ/2c) exp(τb) exp(τ/2c). Example (Leap-Frog method): S 1(τ) = exp(τ/2a) exp(τ/2c) exp(τb) exp(τ/2c) exp(τ/2a). REMARK: this introduces an extra error term in the approximation of order ε 3 τ 3. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

40 Remark 2: Compensated Summation When we solve numerically an ODE, we essentially have a recursive evaluation of the form: y n+1 = y n + δ n, (7) where y n is the approximated solution and δ n is the increment to be done. Usually δ n y n. The evaluation of Eq. (7) can cause larger rounding errors that the computation of δ n. To reduce this round-off error we can use the so called the compensated summation algorithm introduced by Kahan Kahan W., 1965: Pracniques: further remarks on reducing truncation errors Communications of the ACM 8(1) pp also see: summation algorithm. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

41 Remark 2: Compensated Summation Definition (Compensated Summation Algorithm) Let y 0 and {δ n } n 0 be given and assume that we want to compute the terms y n+1 = y n + δ n. We start with e = 0 and compute y 1, y 2,... as follows: for n = 0, 1, 2,... do a = y n e = e + δ n y n+1 = a + e e = e + (a y n+1 ) enddo Notice that with this algorithm is to accumulate the rounding errors in e and feed them back into the summation when possible. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

42 Remark 2: Compensated Summation The CODE would look something like this: subroutine pas B(Xplan,XPplan,tau) implicit none integer i real(treal),intent(in) :: tau real(treal),dimension(3,nplan),intent(inout):: real(treal), dimension(3):: AUX call Accelera(Xplan) do i=1,nplan AUX = XPplan(:,i) err(4:6,i,1) = err(4:6,i,1) - tau*(cg*acc(:,i)) XPplan(:,i) = AUX + err(4:6,i,1) err(4:6,i,1) = err(4:6,i,1) + (AUX - XPplan(:,i)) end do end subroutine pas B Xplan,XPplan A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

43 Remark 2: Compensated Summation Results: Compensated Summation Vs No Compensated Summation Jup - Sat [ Double Precision - CS vs no CS ] Jup - Sat [ Extended Precision - CS vs no CS ] SABA1 (CS) SABA2 (CS) SABA3 (CS) SABA4 (CS) SABA1 (no CS) SABA2 (no CS) SABA3 (no CS) SABA4 (no CS) SABA1 (CS) SABA2 (CS) SABA3 (CS) SABA4 (CS) SABA1 (no CS) SABA2 (no CS) SABA3 (no CS) SABA4 (no CS) Figure: Maximum variation of the energy versus cost of the SABA n schemes on the Sun - Jupiter - Saturn three body problem, with and without the compensated summation. Results using double precision arithmetics (left) and extended precision arithmetics (right). A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

44 End of REMARKS!!

45 Jacobi vs Heliocentric (first results) Jupiter - Saturn - Uranus - Neptune (Jacobi Coord) SABA4 (8,2) SABA4 m=2 (8,4) SABAC4 (8,4) McLa ABA (8,4) Jupiter - Saturn - Uranus - Neptune (Helio Coord) SABAH4 (8,2) SABAH4 m=2 (8,4) McLa ABAH (8,4) Bla ABAH (8,4,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

46 More Splitting Methods (work with S. Blanes et.al) Our goal is to find new splitting symplectic schemes that will improve the results already discussed. Improve the performance of McLachlan in Heliocentric Coordinates. Build new schemes for Jacobi Coordinates. Build new schemes for Heliocentric Coordinates. Compare the performance of all of these schemes trying to find the optimal one for our purpose. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

47 Heliocentric Coordinates (Improving McLachlan) As we have already discussed, in Heliocentric coordinates, we use exp(τ/2c) exp(τb) exp(τ/2c) to integrate the perturbation part. This introduces in our approximation error terms of order ε 3 τ 2 that can become important for small step-sizes. For instance, the McLachlan methods of order (8, 4) becomes a method of order (8, 4, 2) In order to improve the performance of these scheme, we can add an extra stage to get rid of these term. m+1 exp(a i τa) exp(b i ετb) i=1 We must add the extra condition: b b b 3 m = 0 A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

48 Heliocentric Coordinates (Improving McLachlan) id order n a i b i ABAH84 (8, 4) 5 BAB84 (8, 4) 5 ABAH844 (8, 4, 4) 6 BABH844 (8, 4, 4) 6 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = a1 = a2 = a3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b1 = b2 = b3 = b4 = A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

49 Heliocentric Coordinates (Improving McLachlan) Mercury - Venus - Earth - Mars (Helio Coord) SABAH4 (8,2) SABAH4 m=2 (8,4) McLa ABAH (8,4) Bla ABAH (8,4,4) Jupiter - Saturn - Uranus - Neptune (Helio Coord) SABAH4 (8,2) SABAH4 m=2 (8,4) McLa ABAH (8,4) Bla ABAH (8,4,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

50 New Schemes In this philosophy, we can always add extra stages in order to kill the desired terms in the error approximation. We need: S m(τ) = m exp(a i τa) exp(b i ετb) i=1 First to decide which are the most relevant terms that might be limiting our splitting scheme. Find the minimal set of coefficients that fulfil our requirements (not trivial). Possible drawbacks: Sometimes many stages are required having no actual gain in the performance of the scheme. We will no longer have positive coefficients. This can sometimes produce big rounding error propagation for long term-integration. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

51 New Schemes for Jacobi Coordinates id order n a i b i a 1 = b ABA82 (8, 2) 4 a 2 = = b a 3 = = ABA84 (8, 4) 5 ABA104 (10, 4) 7 ABA864 (8, 6, 4) 7 ABA864 eo(10, 8, 6) (8, 6, 4) 9 ABA1064 (10, 6, 4) 8 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 5 = 0.5 (a 1 + a 2 + a 3 + a 4) a 1 = a 2 = a 3 = a 4 = a 5 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = b 5 = 1 2(b 1 + b 2 + b 3 + b 4) b 1 = b 2 = b 3 = b 4 = A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

52 New Schemes for Heliocentric Coordinates id order n a i b i a 1 = b ABAH82 (8, 2) 4 a 2 = = b a 3 = = ABAH84 (8, 4) 5 ABAH844 (8, 4, 4) 6 ABAH864 (8, 6, 4) 8 ABAH1064 (10, 6, 4) 9 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 5 = a 1 = a 2 = a 3 = a 4 = a 5 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = b 5 = A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

53 Results for Jacobi (I) Mer - Ven - Ear - Mar (Jacobi Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bl ABA (10,4) Bl ABA (8,6,4)* Bl ABA (8,6,4) Bl ABA (10,6,4) Jup - Sat - Ura - Nep (Jacobi Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bl ABA (10,4) Bl ABA (8,6,4)* Bl ABA (8,6,4) Bl ABA (10,6,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

54 Results for Jacobi (II) Mercury to Neptune (Jacobi Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bl ABA (10,4) Bl ABA (8,6,4)* Bl ABA (8,6,4) Bl ABA (10,6,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

55 Results for Heliocentric (I) Mer - Ven - Ear - Mar (Helio Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bla ABA (8,4,4) Bla ABA (8,6,4) Bla BAB (8,6,4) Bla ABA (10,6,4) Jup - Sat - Ura - Nep (Helio Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bla ABA (8,4,4) Bla ABA (8,6,4) Bla BAB (8,6,4) Bla ABA (10,6,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

56 Results for Heliocentric (II) Mercury to Neptune (Helio Coord) [ short ] La SABA4 (8,2) McLa ABA (8,4) Bla ABA (8,4,4) Bla ABA (8,6,4) Bla BAB (8,6,4) Bla ABA (10,6,4) A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

57 Results Jacobi Vs Heliocentric (I) Mer-Ven-Earth-Mars (Helio vs Jacobi) [ long ] Helio ABAH844 Helio ABAH1064 Jacob ABA84 Jacob ABA1064 Jacob ABA Jup-Sat-Ura-Nep (Helio vs Jacobi) [ long ] Helio ABAH844 Helio ABAH1064 Jacob ABA84 Jacob ABA1064 Jacob ABA A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

58 Results Jacobi Vs Heliocentric (II) Mercury to Neptune (Helio vs Jacobi) [ long ] Helio ABAH844 Helio ABAH1064 Jacob ABA84 Jacob ABA1064 Jacob ABA A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

59 Final Comments Jacobi coordinates offer better results than Heliocentric coordinates. The use of a corrector is needed in order to improve the efficiency of the splitting methods. Adding extra stages in order to improve the error approximation (i.e. methods of order (8, 4, 4), (8, 6, 4),... ) in many cases improves the results. The high angular momenta of Mercury is the main limiting factor on the optimal step-size. A. Farrés (IMCCE) SympMeth for Long-Term Integration of SolSys Abril 22nd, / 60

60 Thank You for Your Attention