Geometric Numerical Integration TU Müncen Ernst Hairer January February 010 Lecture : Symplectic integrators Table of contents 1 Basic symplectic integration scemes 1 Symplectic Runge Kutta metods 4 3 Te Adjoint of a Metod 7 4 Composition metods 8 5 Splitting metods 1 6 Integrators based on generating functions 15 7 Variational integrators 15 8 Exercises 18 A numerical one-step metod y n+1 = Φ (y n ) is called symplectic if, wen applied to a Hamiltonian system, te discrete flow y Φ (y) is a symplectic transformation for all sufficiently small step sizes. Pioneering work on symplectic integrators is due to de Vogelaere (1956) 1, Rut (1983), and Feng Kang (1985) 3. 1 Basic symplectic integration scemes Te most simple symplectic integrators are motivated by te teory of generating functions for symplectic transformations (see Lecture 1). We consider te Hamiltonian system in te variables y = (p, q), ṗ = q H(p, q) q = p H(p, q) or equivalently ẏ = J 1 H(y). 1 R. de Vogelaere, Metods of integration wic preserve te contact transformation property of te Hamiltonian equations, Report No. 4, Dept. Mat., Univ. of Notre Dame, Notre Dame, Ind. (1956) R.D. Rut, A canonical integration tecnique, IEEE Trans. Nuclear Science NS-30 (1983) 669 671. 3 K. Feng, On difference scemes and symplectic geometry, Proceedings of te 5-t Intern. Symposium on differential geometry & differential equations, Aug. 1984, Beijing (1985) 4 58. 1
Teorem 1 (symplectic Euler) Te so-called symplectic Euler metods p n+1 = p n q H(p n+1, q n ) q n+1 = q n + p H(p n+1, q n ) are symplectic metods of order 1. or p n+1 = p n q H(p n, q n+1 ) q n+1 = q n + p H(p n, q n+1 ) Proof. Symplecticity is an immediate consequence of te first two caracterizations of Teorem 5 (Lecture 1). Consistency of te sceme is obvious. Te metods (1) are implicit for general Hamiltonian systems. For separable H(p, q) = T(p)+U(q), owever, bot variants turn out to be explicit. Tis is also te case for te metods of te next teorem. Teorem (Störmer Verlet) Te Störmer Verlet scemes p n+1/ = p n qh(p n+1/, q n ) q n+1 = q n + ( ) p H(p n+1/, q n ) + p H(p n+1/, q n+1 ) (1) () and p n+1 = p n+1/ qh(p n+1/, q n+1 ) q n+1/ = q n + ph(p n, q n+1/ ) p n+1 = p n ( ) q H(p n, q n+1/ ) + q H(p n+1, q n+1/ ) (3) q n+1 = q n+1/ + ph(p n+1, q n+1/ ) are symplectic metods of order. Proof. Te statement follows from te fact tat te Störmer Verlet sceme is te composition of te two symplectic Euler metods (1) wit step size /. Even order follows from its symmetry. For a second order differential equation q = U(q), for wic te Hamiltonian is H(p, q) = 1 pt p + U(q), metod () becomes q n+1 q n + q n 1 = U(q n ), p n = q n+1 q n 1. (4) Teorem 3 (implicit midpoint) Te implicit midpoint rule ( y n+1 = y n + J 1 yn+1 + y ) n H is a symplectic metod of order. (5)
0 4 6 8 0 4 6 8 explicit Euler Runge, order 0 4 6 8 symplectic Euler 0 4 6 8 Verlet 0 4 6 8 implicit Euler 0 4 6 8 midpoint rule Figure 1: Area preservation of numerical metods for te pendulum; same initial sets as in Figure 3 of Lecture 1; first order metods (left column): = π/4; second order metods (rigt column): = π/3; dased: exact flow. Proof. Symplecticity is a consequence of te tird caracterization of Teorem 5 (Lecture 1), and order follows from te symmetry of te metod. We consider te pendulum problem wit Hamiltonian H(p, q) = 1 p cosq. We apply six different numerical metods to tis problem: te explicit Euler metod, te symplectic Euler metod (1), and te implicit Euler metod, as well as a second order metod of Runge, te Störmer Verlet sceme (), and te implicit midpoint rule (5). For two sets of initial values (p 0, q 0 ) we compute several steps wit step size = π/4 for te first order metods, and = π/3 for te second order metods. One clearly observes in Figure 1 4 tat te explicit Euler, te implicit Euler and te second order explicit metod of Runge are not symplectic (not area preserving). 4 Tis figure and most of te text are taken from te monograp Geometric Numerical Integration by Hairer, Lubic & Wanner. 3
Symplectic Runge Kutta metods An s-stage Runge Kutta metod, applied to an initial value problem ẏ = f(t, y), y(t 0 ) = y 0 is given by te formulas k i y 1 ( = f t 0 + c i, y 0 + s = y 0 + b i k i, i=1 s a ij k j ), i = 1,...,s j=1 (6) were c i = s j=1 a ij. We allow a full matrix (a ij ) of non-zero coefficients, so tat te slopes k i are defined implicitly. For te study of teir symplecticity we follow te approac of Bocev & Scovel 5. It relates symplecticity to te conservation of quadratic first integrals. Recall tat a function I(y) is a first integral of te differential equation ẏ = f(y), if I (y)f(y) = 0 for all y. Teorem 4 If a Runge Kutta metod (6) conserves quadratic first integrals (i.e., I(y 1 ) = I(y 0 ) wenever I(y) = y T Cy, wit symmetric matrix C, is a first integral of ẏ = f(y)), ten it is symplectic. Proof. Symplecticity of te discrete flow y Φ (y) means tat for a Hamiltonian problem te quadratic expression Ψ T J Ψ (wit Ψ = Φ (y)) is a first integral of te variational equation. Te statement of te teorem is terefore a consequence of te fact tat for Runge Kutta metods te following diagram commutes: ẏ = f(y), y(0) = y 0 metod ẏ = f(y), y(0) = y 0 Ψ = f (y)ψ, Ψ(0) = I metod {y n } {y n, Ψ n } (orizontal arrows mean a differentiation wit respect to y 0 ). Terefore, te numerical result y n, Ψ n, obtained from applying te metod to te problem augmented by its variational equation, is equal to te numerical solution for ẏ = f(y) augmented by its derivative Ψ n = y n / y 0. 5 P.B. Bocev & C. Scovel, On quadratic invariants and symplectic structure, BIT 34 (1994) 337 345. 4
Te commutativity of te above diagram is proved by implicit differentiation. Let us illustrate tis for te explicit Euler metod y n+1 = y n + f(y n ). We consider y n and y n+1 as functions of y 0, and we differentiate wit respect to y 0 te equation defining te numerical metod. For te Euler metod tis gives y n+1 y 0 = y n y 0 + f (y n ) y n y 0, wic is exactly te relation tat we get from applying te metod to te variational equation. Since y 0 / y 0 = I, we ave y n / y 0 = Ψ n for all n. An elegant proof of symplecticity is possible for Gauss collocation metods. Tey are defined as follows: let c 1,...,c s be te zeros of te sifted Legendre polynomial ds dx s ( x s (1 x) s), and let u(t) be te polynomial of degree s satisfying u(t 0 ) = y 0 u(t 0 + c i ) = f ( t 0 + c i, u(t 0 + c i ) ), i = 1,..., s, ten te numerical solution of te Gauss collocation metod is defined by y 1 = u(t 0 + ). Putting k i = u(t 0 + c i ), and expressing u(t 0 + τ) and by integration also u(t 0 +τ) in terms of te k i, one see tat tese metods are special case of implicit Runge Kutta metods. Te implicit midpoint rule (5) is te special case s = 1 of te Gauss metods. Teorem 5 Te Gauss collocation metods conserve quadratic first integrals and are tus symplectic by Teorem 4. Proof. Let u(t) be te collocation polynomial of te Gauss metods, and assume tat I(y) = y T C y, wit symmetric C, is a first integral of ẏ = f(y). Since d I( u(t) ) = u(t) T C u(t), it follows from u(t dt 0 ) = y 0 and u(t 0 + ) = y 1 tat y T 1 C y 1 y T 0 C y 0 = t0 + (7) t 0 u(t) T C u(t) dt. (8) Te integrand u(t) T C u(t) is a polynomial of degree s 1, wic is integrated witout error by te s-stage Gaussian quadrature formula. It terefore follows from te collocation condition u(t 0 + c i ) T C u(t 0 + c i ) = u(t 0 + c i ) T C f ( u(t 0 + c i ) ) = 0 tat te integral in (8) vanises. 5
Te following criterion on te conservation of quadratic first integrals is due to Cooper 6, tat on te symplecticity as been found independently by Lasagni 7, Sanz-Serna 8, and Suris 9. Teorem 6 If te coefficients of a Runge Kutta metod satisfy b i a ij + b j a ji = b i b j for all i, j = 1,...,s, (9) ten it conserves quadratic first integrals and it is symplectic (by Teorem 4). Proof. Te Runge Kutta relation y 1 = y 0 + s i=1 b ik i yields y T 1 Cy 1 = y T 0 Cy 0 + s b i ki T Cy 0 + i=1 s b j y0 T Ck j + j=1 s b i b j ki T Ck j. (10) i,j=1 We ten write k i = f(y i ) wit Y i = y 0 + s j=1 a ijk j. Te main idea is to compute y 0 from tis relation and to insert it into te central expressions of (10). Tis yields (using te symmetry of C) y T 1 Cy 1 = y T 0 Cy 0 + s i=1 b i Y T i Cf(Y i ) + s (b i b j b i a ij b j a ji ) ki T Ck j. i,j=1 Te condition (9) togeter wit te assumption y T Cf(y) = 0, wic states tat y T Cy is a first integral of ẏ = f(y), imply y T 1 Cy 1 = y T 0 Cy 0. All results of tis section can be extended to partitioned Runge Kutta metods, were te components of p and q in a Hamiltonian system are treated by different Runge-Kutta metods. Te most prominent examples are te symplectic Euler metod (combination of explicit and implicit Euler) and te Störmer Verlet metod (combination of te trapezoidal rule and te implicit midpoint rule). 6 G.J. Cooper, Stability of Runge Kutta metods for trajectory problems, IMA J. Numer. Anal. 7 (1987) 1 13. 7 F.M. Lasagni, Canonical Runge Kutta metods, ZAMP 39 (1988) 95 953. 8 J.M. Sanz-Serna, Runge Kutta scemes for Hamiltonian systems, BIT 8 (1988) 877 883. 9 Y.B. Suris, On te conservation of te symplectic structure in te numerical solution of Hamiltonian systems (in Russian), In: Numerical Solution of Ordinary Differential Equations, ed. S.S. Filippov, Keldys Inst. of Appl. Mat., USSR Academy of Sciences, Moscow, 1988, 148 160. 6
3 Te Adjoint of a Metod Te flow ϕ t of an autonomous differential equation ẏ = f(y) satisfies ϕ 1 t = ϕ t. Tis property is in general not sared by te one-step map Φ of a numerical metod. An illustration is presented in te upper picture of Figure (a), were we see tat te one-step map Φ for te explicit Euler metod is different from te inverse of Φ, wic is te implicit Euler metod. Definition 1 Te adjoint metod Φ of a metod Φ is te inverse map of te original metod wit reversed time step, i.e., Φ := Φ 1 (11) (see Figure (b)). In oter words, y 1 = Φ (y 0) is implicitly defined by te relation Φ (y 1 ) = y 0. A metod for wic Φ = Φ is called symmetric. (a) Φ y 1 (b) (c) y 1 e ϕ (y 0 ) y 0 Φ Φ y 1 Φ Φ Φ Φ Φ y 0 Φ y 0 e Φ Figure : Definition and properties of te adjoint metod Te adjoint metod satisfies te usual properties suc as (Φ ) = Φ and (Φ Ψ ) = Ψ Φ for any two one-step metods Φ and Ψ. Te implicit Euler metod is te adjoint of te explicit Euler metod. Te implicit midpoint rule is symmetric (see te lower picture of Figure (a)), and te trapezoidal rule and te Störmer Verlet metod are also symmetric. Teorem 7 Let ϕ t be te exact flow of ẏ = f(y) and let Φ be a one-step metod of order r satisfying Φ (y 0 ) = ϕ (y 0 ) + C(y 0 ) r+1 + O( r+ ). (1) Te adjoint metod Φ ten as te same order r and we ave Φ (y 0) = ϕ (y 0 ) + ( 1) r C(y 0 ) r+1 + O( p+r ). (13) If te metod is symmetric, its (maximal) order is even. 7
Proof. Te idea of te proof is exibited in drawing (c) of Figure. From a given initial value y 0 we compute ϕ (y 0 ) and y 1 = Φ (y 0), wose difference e is te local error of Φ. Tis error is ten projected back by Φ to become e. We see tat e is te local error of Φ, i.e., by ypotesis (1), e = ( 1) r C(ϕ (y 0 )) r+1 + O( r+ ). (14) Since ϕ (y 0 ) = y 0 + O() and e = (I + O())e, it follows tat e = ( 1) r C(y 0 ) r+1 + O( r+ ) wic proves (13). Te statement for symmetric metods is an immediate consequence of tis result, because Φ = Φ implies C(y 0) = ( 1) r C(y 0 ), and terefore C(y 0 ) can be different from zero only for even r. 4 Composition metods Let Φ be a basic metod and γ 1,...,γ s real numbers. Ten we call its composition wit step sizes γ 1, γ,...,γ s, i.e., Ψ = Φ γs... Φ γ1, (15) te corresponding composition metod (see Figure 3(a)). Te aim is to increase te order wile preserving desirable properties like symplecticity of te basic metod. Teorem 8 Let Φ be a one-step metod of order r. If γ 1 +... + γ s = 1 γ1 r+1 +... + γs r+1 = 0, ten te composition metod (15) is at least of order r + 1. (16) Proof. Te proof is presented in Figure 3(b) for s = 3. It is very similar to te proof of Teorem 7. Starting wit y 0, we let y i = Φ γi (y i 1 ), so tat Ψ (y 0 ) = y s. By ypotesis we ave e i+1 = ϕ γi (y i ) Φ γi (y i ) = C(y i ) γi r+1 r+1 + O( r+ ), and te transported local error satisfies E i = (I + O())e i for all i. Because of y i = y 0 + O() it follows from s i=1 γ i = 1 (consistency requirement) tat ϕ (y 0 ) Ψ (y 0 ) = E 1 +... + E s = C(y 0 )(γ r+1 1 +... + γ r+1 s ) r+1 + O( r+ ) wic sows tat under conditions (16) te O( r+1 )-term vanises. 8
(a) ϕ Σγi (y 0 ) (b) ϕ Σγi (y 0 ) E 1 E Ψ y 0 y 1 y Φ γ1 Φ γ y 3 e 3 =E 3 e Ψ (y 0 ) Φ γ3 e 1 y 0 y Φ γ3 y 1 Φ γ1 Φ γ Figure 3: Composition of metod Φ wit tree step sizes Example 1 (Te Triple Jump) Equations (16) ave no real solution for odd r. Terefore, te order increase is only possible for even r. In tis case, te smallest s wic allows a solution is s = 3. We ten ave some freedom for solving te two equations. If we impose symmetry γ 1 = γ 3, ten we obtain (Creutz & Gocksc 10, Forest 11, Suzuki 1, Yosida 13 ) 1 γ 1 = γ 3 = 1/(r+1), γ = 1/(r+1). (17) 1/(r+1) Tis procedure can be repeated: we start wit a symmetric metod of order, apply (17) wit r = to obtain order 3; due to te symmetry of te γ s tis new metod is in fact of order 4 (see Teorem 7). Wit tis new metod we repeat (17) wit r = 4 and obtain a symmetric 9-stage composition metod of order 6, ten wit r = 6 a 7-stage symmetric composition metod of order 8, and so on. One obtains in tis way any order, owever, at te price of a terrible zig-zag of te step points (see Figure 4). γ p=4 p=6 p=8 3 γ γ 1 1 0 1 1 0 1 1 0 1 Figure 4: Te Triple Jump of order 4 and its iterates of orders 6 and 8 10 M. Creutz & A. Gocksc, Higer-order ybrid Monte Carlo algoritms, Pys. Rev. Lett. 63 (1989) 9 1. 11 E. Forest, Canonical integrators as tracking codes, AIP Conference Proceedings 184 (1989) 1106 1136. 1 M. Suzuki, Fractal decomposition of exponential operators wit applications to many-body teories and Monte Carlo simulations, Pys. Lett. A 146 (1990) 319 33. 13 H. Yosida, Construction of iger order symplectic integrators, Pys. Lett. A 150 (1990) 6 68. 9
Example (Suzuki s Fractals) If one desires metods wit smaller values of γ i, one as to increase s even more. For example, for s = 5 te best solution of (16) as te sign structure + + + + wit γ 1 = γ (Suzuki 1990). Tis leads to 1 γ 1 = γ = γ 4 = γ 5 = 4 4 1/(r+1), γ 3 = 41/(r+1). (18) 4 41/(r+1) Te repetition of tis algoritm for r =, 4, 6,... leads to a fractal structure of te step points (see Figure 5). p=4 γ 4 γ 5 p=6 p=8 γ 3 γ 1 γ 0 1 0 1 0 1 Figure 5: Suzuki s fractal composition metods Composition wit te Adjoint Metod. If we replace te composition (15) by te more general formula Ψ = Φ αs Φ β s... Φ β Φ α 1 Φ β 1, (19) te condition for order r + 1 becomes, by using te result (13) and a similar proof as above, β 1 + α 1 + β +... + β s + α s = 1 ( 1) r β r+1 1 + α r+1 1 + ( 1) r β r+1 +... + ( 1) r β r+1 s + α r+1 s = 0. (0) Tis allows an order increase for odd r as well. In particular, we see at once te solution α 1 = β 1 = 1/ for r = s = 1, wic turns every consistent one-step metod of order 1 into a second-order symmetric metod Ψ = Φ / Φ /. (1) For example, if Φ is te explicit (resp. implicit) Euler metod, ten Ψ in (1) becomes te implicit midpoint (resp. trapezoidal) rule. If Φ is te symplectic Euler metod, ten te composed metod Ψ in (1) is te Störmer Verlet metod. A Numerical Example. To demonstrate te numerical performance of te above metods, we coose te Kepler problem on te interval [0, 7.5] wit initial values corresponding to an eccentricity e = 0.6 (see Lecture 1). As te basic metod we use te Störmer Verlet sceme and compare in Figure 6 te Triple Jump (17) and Suzuki (18) compositions for a large number of different equidistant basic step sizes and for orders r = 4, 6, 8, 10, 1. Te maximal final error is compared wit 10
10 0 10 3 10 6 error 6 4 8 10 1 10 0 4 10 3 10 6 error 6 10 1 8 10 9 10 9 10 1 10 15 Yosida 10 1 10 10 3 10 4 10 5 10 15 8 function eval. 6 Suzuki 10 10 3 10 4 10 5 8 function eval. Figure 6: Numerical results of te Triple Jump and Suzuki step sequences (grey symbols) compared to optimal metods (black symbols) 6 te total number of function evaluations in double logaritmic scales. We observe tat te wild zig-zag of te Triple Jump (17) is a more serious andicap tan te large number of small steps of te Suzuki sequence (18). Optimized composition metods. Te construction of optimal metods (large order wit minimal s) needs an elaborate order teory and cumbersome numerical 14 15 16 searc algoritms. We just present te coefficients of an 8t order metod γ 1 = γ 15 = 0.74167036435061953448780 γ = γ 14 = 0.4091008580003159399730010 γ 3 = γ 13 = 0.19075471096383799538766 γ 4 = γ 1 = 0.57386471116086665638773 γ 5 = γ 11 = 0.990641813036559384446354 γ 6 = γ 10 = 0.33464918459818378495798 γ 7 = γ 9 = 0.31593093967665966305666 γ 8 = 0.7968879393591635401978884 p8 s15 0 1 () Te results of tis 8t order metod and of an optimized metod of order 6 are included in Figure 6 wit black symbols. Tey outperform tose of te previous approaces. 14 M. Suzuki & K. Umeno, Higer-order decomposition teory of exponential operators and its applications to QMC and nonlinear dynamics, Springer Proceedings in Pysics 76 (1993) 74 86. 15 M. Suzuki, Quantum Monte Carlo metods and general decomposition teory of exponential operators and symplectic integrators, Pysica A 05 (1994) 65 79. 16 R.I. McLaclan, On te numerical integration of ordinary differential equations by symmetric composition metods, SIAM J. Sci. Comput. 16 (1995) 151 168. 11
5 Splitting metods We consider an arbitrary system ẏ = f(y) in R n, and suppose tat te vector field is split as (Figure 7) ẏ = f [1] (y) + f [] (y). (3) If te exact flows ϕ [1] t and ϕ [] t of te systems ẏ = f [1] (y) and ẏ = f [] (y) can be calculated explicitly, we can compose tem to get numerical approximations. f = f [1] + f [] Figure 7: A splitting of a vector field. Lie Trotter 17 splitting. If, from a given initial value y 0, we first solve te first system to obtain a value y 1/, and from tis value integrate te second system to obtain y 1, we get numerical integrators Φ = ϕ [] ϕ[1] Φ = ϕ [1] ϕ[] Φ y 1 y 0 y 1/ ϕ [1] ϕ [] y 1/ ϕ [] y 0 ϕ [1] Φ y1 (4) were one is te adjoint of te oter. By Taylor expansion we find tat (ϕ [1] ϕ [] )(y 0) = ϕ (y 0 ) + O( ), so tat bot metods give approximations of order 1 to te solution of (3). Strang 18 splitting or Marcuk 19 splitting. Anoter idea is to use a symmetric version and put Φ [S] ϕ [] ϕ [1] / = ϕ[1] / ϕ[] ϕ[1] /, Φ [S] y 1 (5) By breaking up in (5) ϕ [] = ϕ [] / ϕ[] / y 0 ϕ [1] /, we see tat te Strang splitting 17 H.F. Trotter, On te product of semi-groups of operators, Proc. Am. Mat. Soc.10 (1959) 545 551. 18 G. Strang, On te construction and comparison of difference scemes, SIAM J. Numer. Anal. 5 (1968) 506 517. 19 G. Marcuk, Some applications of splitting-up metods to te solution of matematical pysics problems, Aplikace Matematiky 13 (1968) 103 13. 1
= Φ / Φ / is te composition of te Lie-Trotter metod and its adjoint wit alved step sizes. Te Strang splitting formula is terefore symmetric and of order (see formula (1)). Φ [S] Example 3 (Te Symplectic Euler and te Störmer Verlet Scemes) Suppose we ave a Hamiltonian system wit separable Hamiltonian H(p, q) = T(p) + U(q). We consider tis as te sum of two Hamiltonians, te first one depending only on p, te second one only on q. Te corresponding Hamiltonian systems ṗ = 0 q = p T(p) and ṗ = q U(q) q = 0 (6) can be solved witout problem to yield p(t) = p 0 q(t) = q 0 + t p T(p 0 ) and p(t) = p 0 t q U(q 0 ) q(t) = q 0. (7) Denoting te flows of tese two systems by ϕ T t and ϕ U t, we see tat te symplectic Euler metod (1) is just te composition ϕ T ϕu, and its adjoint is ϕu ϕt. Te Störmer Verlet sceme () is ϕ U / ϕt ϕu /, te Strang splitting (5). General Splitting Procedure. In a similar way to te general idea of composition metods (19), we can form wit arbitrary coefficients a 1, b 1, a,..., a m, b m (were, eventually, a 1 or b m, or bot, are zero) Ψ = ϕ [] b m ϕ[1] a m ϕ[] b m 1... ϕ[1] a ϕ[] b 1 ϕ[1] a 1 (8) and try to increase te order of te sceme by suitably determining te free coefficients. A close connection between te teories of splitting metods (8) and of composition metods (19) was discovered by McLaclan (1995). Indeed, if we put β 1 = a 1 and break up ϕ [] b 1 = ϕ[] α 1 ϕ[] β 1 (group property of te exact flow) were α 1 is given in (30), furter ϕ [1] a = ϕ[1] β ϕ[1] α 1 see, using (4), tat Ψ of (8) is identical wit Ψ of (19), were and so on (cf. Figure 8), we Φ = ϕ [1] ϕ[] so tat Φ = ϕ [] ϕ[1]. (9) A necessary and sufficient condition for te existence of α i and β i satisfying (30) is tat a i = b i, wic is te consistency condition anyway for metod (8). 13
Φ β 3 y 1 ϕ [] b 3 ϕ [] b ϕ [1] Φ a 3 Φ α β ϕ [] b 1 Φ ϕ [1] α1 Φ a β 1 a 1 = β 1 b 1 = β 1 + α 1 a = α 1 + β b = β + α a 3 = α + β 3 b 3 = β 3 (30) y 0 ϕ [1] a 1 Figure 8: Equivalence of splitting and composition metods. Combining Exact and Numerical Flows. If te splitting (3) is suc tat only te flow of, say, ẏ = f [1] (y) can be computed exactly, we can consider Φ = ϕ [1] Φ[], Φ = Φ [] ϕ [1] (31) as te basis of te composition metod (19). Here ϕ [1] t is te exact flow of ẏ = f [1] (y), and Φ [] is some first-order integrator applied to ẏ = f[] (y). Since Φ of (31) is consistent wit (3), te above interpretation of splitting metods as composition metods implies tat te resulting metod Ψ = ϕ [1] α s Φ[] α s Φ[] β s ϕ[1] (β s+α s 1 ) Φ[] α s 1... Φ[] β 1 ϕ[1] β 1 (3) as te desired ig order. Notice tat replacing ϕ [] t wit a low-order approximation Φ [] t in (8) would not retain te ig order of te composition, because Φ [] t does not satisfy te group property. Splitting into More tan Two Vector Fields. Consider a differential equation ẏ = f [1] (y) + f [] (y) +... + f [N] (y), (33) were we assume tat te flows ϕ [j] t of te individual problems ẏ = f [j] (y) can be computed exactly. Tere are many possibilities for extending (8) and for writing te metod as a composition of ϕ [1] a j, ϕ[] b j, ϕ[3] c j,.... A simple and efficient way is to consider te first-order metod Φ = ϕ [1] ϕ[]... ϕ[n] togeter wit its adjoint as te basis of te composition (19). Witout any additional effort tis yields splitting metods for (33) of arbitrary ig order. 14
6 Integrators based on generating functions To construct symplectic numerical metods of ig order, Feng Kang 0 and Cannell & Scovel 1 proposed computing an approximate solution of te Hamilton Jacobi equation. Recall tat a mapping (p n, q n ) (p n+1, q n+1 ) defined by p n+1 = p n q S 1 (p n+1, q n ), q n+1 = q n + p S 1 (p n+1, q n ) (34) is always symplectic (Section 5 of Lecture 1) and tat it reproduces te exact solution (after time ) of te Hamiltonian system (Section 6 of Lecture 1) if were S 1 (p, q, t) = G 1 (p, q) + G (p, q) + 3 G 3 (p, q) +... G 1 (p, q) = H(p, q), G (p, q) = 1 ( H H p q G 3 (p, q) = 1 6 ( H ( H p q If we use te truncated series ) (p, q), ) H H H + p q p q + H ( H ) ) (p, q). q P S 1 (p, q) = G 1 (p, q) + G (p, q) +... + r G r (p, q) (35) and insert it into (34), we obtain a symplectic one-step metod of order r. We remark tat for r te metods obtained require te computation of iger derivatives of H(p, q), and for separable Hamiltonians H(p, q) = T(p) + U(q) tey are no longer explicit (compared to te symplectic Euler metod (1)). 7 Variational integrators All previous approaces start from extremizing te action integral S(q) = tn t 0 L ( q(t), q(t) ) dt, ten deriving te Euler Lagrange equations and te equivalent Hamiltonian equations, and finally discretize te resulting differential equations. 0 K. Feng, Difference scemes for Hamiltonian formalism and symplectic geometry, J. Comp. Mat. 4 (1986) 79 89. 1 P.J. Cannell & J.C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990) 31 59. 15
Variational integrators start from discretizing te action integral followed by extremizing it in a finite dimensional space to obtain discrete Euler Lagrange equations and (symplectic) numerical integrators. For given q 0 and q N, we consider te approximation N 1 S ({q n } N 0 ) = n=0 L (q n, q n+1 ), L (q n, q n+1 ) tn+1 t n L(q(t), q(t)) dt of te action integral, were L plays te role of a discrete Lagrangian. Te requirement S / q n = 0 for an extremum yields te discrete Euler Lagrange equations L y (q n 1, q n ) + L x (q n, q n+1 ) = 0 (36) for n = 1,...,N 1, were te partial derivatives refer to L = L (x, y). Tis gives a tree-term difference sceme for determining q 1,...,q N 1. We introduce te discrete momenta via a discrete Legendre transformation, p n = L x (q n, q n+1 ), (37) so tat te discrete Euler Lagrange equations become equivalent to (substitute n for n + 1 in (36)) p n+1 = L y (q n, q n+1 ). (38) Under suitable assumptions on L, te two equations (37) and (38) define a mapping (p n, q n ) (p n+1, q n+1 ). Teorem 9 Te numerical metod (p n, q n ) (p n+1, q n+1 ), defined by (37) and (38), is a symplectic integrator. Proof. Te differential of L = L (q n, q n+1 ) satisfies dl = p n+1 dq n+1 p n dq n, wic proves symplecticity by Teorem 4 of Lecture 1. Example 4 (MacKay ) Coose L (q n, q n+1 ) by approximating q(t) as te linear interpolant of q n and q n+1 and te integral by te trapezoidal rule. Tis gives L (q n, q n+1 ) = ( L q n, q n+1 q ) n + ( L q n+1, q n+1 q ) n (39) R. MacKay, Some aspects of te dynamics of Hamiltonian systems, in: D.S. Broomead & A. Iserles, eds., Te Dynamics of Numerics and te Numerics of Dynamics, Clarendon Press, Oxford, 199, 137 193. 16
and ence te symplectic sceme, wit v n+1/ = (q n+1 q n )/ for brevity, p n = 1 L q (q n, v n+1/ ) + 1 L q (q n+1, v n+1/ ) L q (q n, v n+1/ ) p n+1 = 1 L q (q n, v n+1/ ) + 1 L q (q n+1, v n+1/ ) + L q (q n+1, v n+1/ ). For a mecanical Lagrangian L(q, q) = 1 qt M q U(q) and wit te notation p n+1/ = Mv n+1/, tis reduces to te Störmer Verlet metod p n+1/ = p n U(q n) q n+1 = q n + M 1 p n+1/ p n+1 = p n+1/ U(q n+1). In tis case, te discrete Euler Lagrange equations (36) become te familiar second order difference formula M(q n+1 q n + q n 1 ) = U(q n ). Example 5 (Wendlandt & Marsden 3 ) Approximating te action integral instead by te midpoint rule gives ( qn+1 + q n L (q n, q n+1 ) = L, q n+1 q ) n. (40) Tis yields te symplectic sceme, wit te abbreviations q n+1/ = (q n+1 +q n )/ and v n+1/ = (q n+1 q n )/, p n = L q (q n+1/, v n+1/ ) L p n+1 = L q (q n+1/, v n+1/ ) + q (q n+1/, v n+1/ ) L q (q n+1/, v n+1/ ). For L(q, q) = 1 qt M q U(q) and p n+1/ = Mv n+1/, tis becomes te implicit midpoint rule p n+1/ = p n U(q n+1/) q n+1 = q n + M 1 p n+1/ p n+1 = p n+1/ U(q n+1/), because we ave p n+1/ = (p n+1 + p n )/. 3 J.M. Wendlandt & J.E. Marsden, Mecanical integrators derived from a discrete variational principle, Pysica D 106 (1997) 3 46. 17
8 Exercises 1. Prove tat under te condition (9) a Runge Kutta metod preserves all first integrals of te form I(y) = y T Cy + d T y + c.. Prove tat te Gauss metods of maximal oder s are te only collocation metods satisfying (9). Hint. Use te ideas of te proof of Lemma 13.9 in Hairer & Wanner 4. 3. Sow tat eac of te symplectic Euler metods in (1) is te adjoint of te oter. 4. Consider te composition metod (15) wit s = 5, γ 5 = γ 1, and γ 4 = γ. Among te solutions of γ 1 + γ + γ 3 = 1, γ 3 1 + γ3 + γ3 3 = 0 find te one tat minimizes γ 5 1 + γ5 + γ5 3. Remark. Tis property motivates te coice of te γ i in (18). 5. Design a symmetric splitting metod for te Euler equations of a rigid body wit given principal momenta of inertia I 1, I, I 3 0 y 3 y ẏ = B(y) H(y), B(y) = y 3 0 y 1 y y 1 0 by splitting te Hamiltonian into tree parts. H(y 1, y, y 3 ) = 1 ( y 1 I 1 + y I + y 3 I 3 ) 4 E. Hairer & G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential- Algebraic Problems, nd edition, Springer Series in Computational Matematics 14, Springer- Verlag Berlin, 1996. 18