Linear energy-preserving integrators for Poisson systems

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1 BIT manucript No. (will be inerted by the editor Linear energy-preerving integrator for Poion ytem David Cohen Ernt Hairer Received: date / Accepted: date Abtract For Hamiltonian ytem with non-canonical tructure matrix a new cla of numerical integrator i propoed. The method exactly preerve energy, are invariant with repect to linear tranformation, and have arbitrarily high order. Thoe of optimal order alo preerve quadratic Caimir function. The dicuion of the order i baed on an interpretation a partitioned Runge Kutta method with infinitely many tage. Keyword Poion ytem energy preervation Caimir function partitioned Runge Kutta method collocation Gauian quadrature PACS 45.2.Jj 2.3.Hq 2.6.Lj Mathematic Subject Claification (2 65P1 65L6 1 Introduction We conider non-canonical Hamiltonian ytem ẏ = B(y H(y, y(t = y, (1.1 where B(y i a kew-ymmetric matrix, o that the energy H(y i preerved along exact olution of (1.1. In thi article we do not require that B(y atifie the Jacobi identity. Our main interet i the deign of numerical integrator that exactly preerve the Hamiltonian H(y and are invariant with repect to linear tranformation. Thi work wa partially upported by the Fond National Suie, project No D. Cohen Mathematiche Intitut, Univerität Bael, CH-451 Bael, Switzerland. David.Cohen@uniba.ch E. Hairer Section de Mathématique, Univerité de Genève, CH-1211 Genève 4, Switzerland. Ernt.Hairer@unige.ch

2 2 David Cohen, Ernt Hairer There i a lot of reearch activity in energy preerving numerical integrator, and variou dicrete gradient method have been propoed in the literature, ee [2,8]. For thee method the numerical olution i typically not invariant with repect to arbitrary linear tranformation. More recently, linear energy preerving integrator have been propoed for canonical Hamiltonian ytem. The averaged vector field method [8, 9] extend the implicit mid-point rule and require the accurate computation of integral. The article [7] preent Runge Kutta method that preerve the energy for polynomial Hamiltonian. An extenion of the averaged vector field integrator to arbitrarily high order i propoed and analyzed in [4]. Until now, energy preerving integrator for non-canonical Hamiltonian ytem that are invariant with repect to linear tranformation, do not eem to exit. The preent article i devoted to an extenion of the method introduced in [4] to problem of the form (1.1. The mot imple example i ( y + y 1 y 1 = y + hb H ( y + τ(y 1 y dτ (1.2 2 which reduce to the averaged vector field integrator for the cae where B(y i a contant matrix. Note that thi method treat the factor B(y and H(y in (1.1 differently, and hould thu be conidered a a partitioned method. A will be hown, thi cheme exactly preerve energy and quadratic Caimir function, it i ymmetric and of order 2. The new cla of method i preented in Section 2. It i a variant of claical collocation method. Section 3 tudie and prove propertie uch a exact preervation of energy and Caimir function, ymmetry, and invariance with repect to linear tranformation. An interpretation of the propoed method a partitioned Runge Kutta method i given in Section 4. Thi allow one to get information on the correct order of convergence. Numerical experiment are preented in Section 5. 2 Energy-preerving integration method We preent a cla of numerical time integrator that exactly preerve the energy H(y, are invariant with repect to linear coordinate tranformation, and are of arbitrarily high order. 2.1 Definition of the numerical integrator Motivated by claical collocation method, we tart by conidering an -point quadrature formula with node c i. The correponding weight b i can be obtained from the Lagrange bai polynomial in interpolation a follow: l i (τ =, j i τ c j, b i = c i c j l i (τdτ.

3 Linear energy-preerving integrator for Poion ytem 3 Definition 2.1 (Energy-preerving integrator Let c 1,...,c be ditinct real number (uually c i 1 for which b i for all i. We conider a polynomial u(t of degree atifying u(t = y (2.1 u(t + c j h = B ( u(t + c j h l j (τ H ( u(t + τh dτ. (2.2 The numerical olution after one tep i then defined by y 1 = u(t + h. Note that approximating the integral by the quadrature formula with node c i and weight b i reduce the integrator to a claical collocation method. For quadratic Hamiltonian the integrand in (2.2 i polynomial of degree 2 1, o that for Gau point c i (zero of hifted Legendre polynomial claical collocation i obtained for general B(y. Thi integrator treat the argument in B(y and in H(y differently and can thu be conidered a a partitioned numerical method. The numerical olution therefore depend on the particular factorization of the vector field. If B(y = B i a contant matrix (e.g., if (1.1 i a canonical Hamiltonian ytem, the method become the energy-preerving integrator of [4]. 2.2 Implementation iue If we denote Y τ := u(t +τh and, with abue of notation, Y j := u(t +c j h, Lagrange interpolation how that u(t + τh = and by integration we arrive at Y τ = y + h l j (σ l j (τb(y j H ( Y σ dσ, (2.3 ( l j (σ τ l j (αdα B(Y j H ( Y σ dσ. (2.4 The polynomial u(t of degree can be expreed in term of y and Y 1,...,Y. Therefore, we need equation (2.4 only for τ = c i (i = 1,..., to compute thi polynomial. Thi repreent a nonlinear ytem of equation for the unknown Y 1,...,Y which can be olved by iteration. The complexity i imilar to that for implicit Runge Kutta method with tage, and identical to that of the energy-preerving integrator of [4]. 2.3 Example Method of optimal order will be obtained when c 1,...,c are the zero of the th Legendre polynomial. The correponding quadrature formula (Gau formula are of order r = 2. Cae = 1. We have c 1 = 1/2 and obtain the method (1.2. It i ymmetric, of order two, and i an extenion of the implicit mid-point rule.

4 4 David Cohen, Ernt Hairer Cae = 2. The node of the Gau quadrature are c 1,2 = 1/2 3/6. With l 1 (τ = (τ c 2 /(c 1 c 2 and l 2 (τ = (τ c 1 /(c 2 c 1 the method read Y 1 =y + h Y 2 =y + h y 1 =y + h ( 1 ( l 1(σB(Y l 2 (σb(y 2 H(Y σ dσ 3 ( ( l 1 (σb(y l 2(σB(Y 2 H(Y σ dσ ( l 1 (σb(y 1 + l 2 (σb(y 2 H(Y σ dσ, where Y σ i the polynomial of degree 2 that interpolate the value y,y 1,Y 2 at t,t + c 1 h,t + c 2 h, repectively. A we hall ee in the next ection, thi method exactly conerve the energy and quadratic Caimir function, it i ymmetric, of order 4, and it i invariant with repect to linear tranformation. 3 Propertie of the new cla of integrator The method of the previou ection have been deigned to have exact energy preervation. It turn out that they have further intereting propertie. 3.1 Exact energy preervation Theorem 3.1 If B(y i kew-ymmetric for all y, then the numerical method of Definition 2.1 exactly preerve the energy, i.e., H(y n = Cont. Proof From the fundamental theorem of calculu we have H ( u(t + h H ( u(t = h H ( u(t + τh T u(t + τhdτ. Replacing the derivative of u(t by (2.3 thi expreion become ( l j (τ H ( u(t + τh T dτ B(Y j l j (σ H ( u(t + σh dσ which vanihe by the kew-ymmetry of the matrix B(y. 3.2 Conervation of quadratic Caimir A function C(y i called a Caimir function of the differential equation (1.1 if C(y T B(y = for all y. Along olution of (1.1 we have C ( y(t = Cont, becaue d dt C( y(t = C ( y(t T B ( y(t H ( y(t =. Thi property i independent of the Hamiltonian H(y.

5 Linear energy-preerving integrator for Poion ytem 5 Theorem 3.2 Let C(y = y T Ay (with a ymmetric contant matrix A be a Caimir function of the ytem (1.1. The energy preerving method baed on the Gauian quadrature formula of order 2 exactly preerve thi Caimir. Proof Uing again the fundamental theorem of calculu we have C ( u(t + h C ( u(t = h C ( u(t + τh T u(t + τhdτ. Since the integrand i a polynomial of degree 2 1, an application of the Gauian quadrature (,c j give the exact reult. The difference C(y 1 C(y i thu equal to h C ( u(t + c j h T B ( u(t + c j h which vanihe and prove the tatement. l j (τ H ( u(t + τh dτ, The equation of motion for a free rigid body are a Lie-Poion ytem with Caimir C(y = y 2 2. The method baed on Gauian quadrature thu preerve exactly the Hamiltonian and the Caimir. 3.3 Symmetry of the integrator Theorem 3.3 If the node c i are ymmetric, i.e., c +1 i = 1 c i, then the numerical method of Definition 2.1 i ymmetric. Proof By the ymmetry of the node we have l +1 i (τ = l i (1 τ. Thi implie that the method, written a y 1 = Φ h (y atifie Φ h (y = Φh 1 (y. 3.4 Invariance with repect to linear tranformation A linear change of coordinate z = Ty tranform a differential equation ẏ = f (y into ż = f (z with f (z = T f (T 1 z. A numerical method Φ f h (y i called linear, if for any T we have T Φ f h (y = Φ f h (Ty. (3.1 For the differential equation (1.1 the tranformed problem i ż = B(z Ĥ(z, z(t = z (3.2 with B(z = T B(T 1 zt T and Ĥ(z = H(T 1 z. We call the integrator of Definition 2.1 linear, if (3.1 hold for f (z = B(z Ĥ(z with thi pecial factorization of the vector field. 1 1 Since the numerical olution of our method depend on the factorization B(y H(y of the vector field, it i important to fix the factorization in the definition of linearity. There i a light abue of notation by calling thi weaker property till linear, but the ame i done for partitioned linear multitep method, where the partitioning of the ytem mut alo be fixed.

6 6 David Cohen, Ernt Hairer Theorem 3.4 The integrator of Definition 2.1 i linear, i.e., if {y n } and {z n } are the numerical olution correponding to (1.1 and (3.2, repectively, and if z = Ty, then we have z n = Ty n for all n. Proof Only expreion of the form B(y H(ỹ with poibly different y and ỹ are involved in the definition of the method. The differential equation (1.1 i aid to have a linear ymmetry T, if with y(t alo z(t = Ty(t i a olution. Thi i equivalent to T B(T 1 z H(T 1 z = B(z H(z, i.e., the tranformed problem (3.2 i equal to the original one. Corollary 3.1 Aume that the problem (1.1 poee a linear ymmetry T that atifie T B(T 1 z H(T 1 z = B(z H( z for all z and z. Then, the dicrete flow of the method of Definition 2.1 ha T a ymmetry, i.e., if {y n } i a olution, then {Ty n } i alo a olution. We remark that the condition on the linear ymmetry in Corollary 3.1 i lightly weaker than what i called linear pecial ymmetry in Propoition 3.8 of [8]. 4 Dicuion of the order If the quadrature formula (b i,c i i of order r = + k ( k, then the order of the method of Definition 2.1 i at leat k + 1. Thi follow at once from b 1 i l i(τ H ( u(t + τh dτ = H ( u(t + c i h + O(h k+1, becaue the method i a perturbation of the claical collocation method. But what i the correct order? The anwer can be obtained by interpreting the method a a partitioned Runge Kutta method. 4.1 Extenion of a theorem of Butcher For a partitioned ytem of differential equation ẏ= f (y,z, y(t = y ż=g(y,z, z(t = z (4.1 we conider method that treat the y an z variable by different Runge Kutta cheme. They are defined by (for i = 1,..., Y i = y + h y 1 = y + h a i j f (Y j,z j b i f (Y i,z i Z i = z + h z 1 = z + h â i j g(y j,z j bi g(y i,z i. (4.2 Order condition, which are in one-to-one correpondence with bi-coloured rooted tree, are dicued in [6, Sect. II.15] and [5, Sect. III.2.2]. Elegant ufficient condition for order p are baed on implifying aumption. For convenience of notation,

7 Linear energy-preerving integrator for Poion ytem 7 we aume b i = b i for all i, and we let c i = a i j and ĉ i = âi j. We then conider the condition B(ρ : C(η : D(ζ : b i ci k 1 ĉi l = 1 k + l, 1 k + l ρ a i j c k 1 j ĉ j l = 1 k + l ck+l i, 1 k + l η b i ci k 1 ĉi l a i j = k+l (1 ĉ j, 1 k + l ζ k + l and we let Ĉ(η, D(ζ be a C(η,D(ζ with a i j replaced by â i j, and, only for Ĉ(η, we replace alo c i in the right-hand ide by ĉ i. Theorem 4.1 If a partitioned Runge Kutta method (4.2 with b i = b i for all i atifie B(ρ, C(η, Ĉ(η, D(ζ, D(ζ, then it i at leat of order p = min(ρ,2η + 2,ζ + η + 1. The proof i the ame a for (non-partitioned Runge Kutta method (ee [6, p. 28]. The only difference i that bi-coloured tree have to be conidered intead of tree. 4.2 Interpreting the energy-preerving integrator a a partitioned RK-method The energy-preerving integrator of Definition 2.1 treat the variable y in B(y and in H(y not in the ame way. We therefore ue different letter and put Y iτ := Y τ and Z iτ := Y i, o that formula (2.4 can be written a with z = y and a iτ, jσ = l j(σ Y iτ =y + h Z iτ =z + h τ l j (αdα, a iτ, jσ B(Z jσ H ( Y jσ dσ â iτ, jσ B(Z jσ H ( Y jσ dσ (4.3 â iτ, jσ = l j(σ For the numerical approximation after one tep we get where y 1 =y + h z 1 =z + h b iτ B(Z iτ H ( Y iτ dτ ci l j (αdα. (4.4 biτ B(Z iτ H ( Y iτ dτ, (4.5 b iτ = b iτ = l i (τ. (4.6

8 8 David Cohen, Ernt Hairer We notice that the formula (4.3-(4.5 contitute a partitioned Runge Kutta method that ha two particularitie. Firt of all, it i conitent with the partitioned ytem of differential equation ẏ=b(z H(y, y(t = y ż=b(z H(y, z(t = z. For z = y we have z(t = y(t for all t and both olution component are equal to the olution of (1.1. Secondly, we are concerned with a partitioned Runge Kutta method having infinitely many tage. Wherea the tage are indexed by i {1,...,} in the method (4.2, they are indexed by (i,τ {1,...,} [,1] in the method (4.3-(4.5. A um over i in (4.2 correpond to a um over i and an integral over τ in the method (4.3-( Verification of the implifying aumption For the coefficient (4.4 we have, uing l j(τ = 1, c iτ = a iτ, jσ dσ = τ, ĉ iτ = We are now ready to check the implifying aumption. â iτ, jσ dσ = c i. (4.7 Lemma 4.1 Let (b i,c i repreent a quadrature formula of order r. The coefficient (4.4, (4.6, and (4.7 then atify B(ρ : C(η : Ĉ(η : D(ζ : D(ζ : b iτ ciτ k 1 ĉiτ l dτ = 1 k + l, 1 k + l ρ a iτ, jσ c k 1 jσ ĉ jσ l dσ = 1 k + l ck+l iτ, 1 k + l η â iτ, jσ c k 1 jσ ĉ jσ l dσ = 1 k + l ĉ iτ k+l, 1 k + l η b iτ ciτ k 1 ĉiτ l a iτ, jσ dτ = σ k+l (1 ĉ jσ k + l, 1 k + l ζ b iτ ciτ k 1 ĉiτ l âiτ, jσ dτ = σ k+l (1 ĉ jσ k + l, 1 k + l ζ with ρ = r, η = min(,r + 1, and ζ = min( 1,r. Proof The proof i baed on the identitie g(σdσ = b ig(c i for polynomial of degree r 1, and l i(τg(c i = g(τ for polynomial of degree 1. We preent detail for the implifying aumption D(ζ. The other condition are proved in a imilar way.

9 Linear energy-preerving integrator for Poion ytem 9 Inerting the expreion (4.4, (4.6 and (4.7 into the left-hand ide of condition D(ζ and uing l i(τc l i = τl for l 1 yield l i (ττ k 1 c l i l j (σ τ l j (αdα dτ = τ k+l 1 l j(σ Partial integration then how that thi expreion equal... = l ( j(σ τ k+l τ l j (αdα τ=1 k + l τ k+l τ= k + l l j(τdτ τ = l ( j(σ k + l l j (αdα dτ. 1 c k+l j which correpond to the right-hand ide of D(ζ. In the lat equality we have replaced the integral over τ by the quadrature formula, which doe not introduce an error when k + l + r. Since k 1, the implifying aumption D(ζ thu hold with ζ = min( 1,r. Standard limit conideration how that Theorem 4.1 i alo valid for partitioned Runge Kutta method with a continuum of tage. We therefore have proved the following reult. Theorem 4.2 Let (b i,c i repreent a quadrature formula of order r. The energypreerving integrator of Definition 2.1 ha order p = min(r,2r For method baed on Gauian quadrature we have order p = 2. An order reduction (compared to claical collocation method occur only for r < 2 2., 5 Numerical experiment After having hown many nice and important feature for the new cla of energypreerving integrator, it i natural to invetigate the quetion to which extent the propoed method are cloe to Poion integrator. It i obviou that they cannot conerve the Hamiltonian and the Poion tructure at the ame time, becaue thi i not poible for the pecial cae of canonical Hamiltonian ytem (ee [1]. But, i it poible that our energy-preerving integrator are conjugate to a Poion integrator? Thi would imply that all Caimir function are nearly conerved without drift. To tudy thi quetion, we conider a Lotka Volterra ytem [3], for which B(y = cy 1y 2 bcy 1 y 3 cy 1 y 2 y 2 y 3, H(y = aby 1 + y 2 ay 3 + ν lny 2 µ lny 3, bcy 1 y 3 y 2 y 3 and where the parameter atify abc = 1. The correponding Poion ytem poee the Caimir C(y = ablny 1 blny 2 + lny 3. For our experiment we have choen a parameter a = 2, b = 1, c =.5, ν = 1, µ = 2, and initial value y( = (1.,1.9,.5. With thee data the olution i periodic.

10 1 David Cohen, Ernt Hairer Fig. 5.1 Time evolution of the error in the Caimir function along the numerical olution of the fourth order energy-preerving method baed on Gau point. To thi problem we apply the fourth order integrator of Section 2.3 with tep ize h =.1. By contruction, the Hamiltonian H(y i preerved up to round-off along the numerical olution. However, the error in the Caimir function C(y how a linear drift of ize O(th 4, ee Figure 5.1. Thi demontrate that the integrator i not conjugate to a Poion integrator. 6 Concluion In thi work we propoe a new cla of energy-preerving integrator for Hamiltonian ytem with non-canonical tructure matrix. The method conerve alo quadratic Caimir function and they are of arbitrarily high order. In contrat to previouly propoed energy-preerving method, our method are invariant with repect to linear tranformation. Reference 1. Chartier, P., Faou, E., Murua, A.: An algebraic approach to invariant preerving integrator: the cae of quadratic and Hamiltonian invariant. Numer. Math. 13, (26 2. Gonzalez, O.: Time integration and dicrete Hamiltonian ytem. J. Nonlinear Sci. 6, ( Grammatico, B., Ollagnier, J.M., Ramani, A., Strelcyn, J.M., Wojciechowki, S.: Integral of quadratic ordinary differential equation in R 3 : the Lotka-Volterra ytem. Phy. A 163(2, ( Hairer, E.: Energy-preerving variant of collocation method. JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, (21 5. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preerving Algorithm for Ordinary Differential Equation, 2nd edn. Springer Serie in Computational Mathematic 31. Springer-Verlag, Berlin (26 6. Hairer, E., Nørett, S.P., Wanner, G.: Solving Ordinary Differential Equation I. Nontiff Problem, 2nd edn. Springer Serie in Computational Mathematic 8. Springer, Berlin ( Iavernaro, F., Trigiante, D.: High-order ymmetric cheme for the energy conervation of polynomial Hamiltonian problem. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4(1-2, (29 8. McLachlan, R.I., Quipel, G.R.W., Robidoux, N.: Geometric integration uing dicrete gradient. Philo. Tran. R. Soc. Lond. Ser. A 357, ( Quipel, G.R.W., McLaren, D.I.: A new cla of energy-preerving numerical integration method. J. Phy. A 41(4, 45,26, 7 (28