A combinatorial approach to integrals of Kahan-Hirota-Kimura discretizations
aa r X i v : . [ n li n . S I] N ov A COMBINATORIAL APPROACH TO INTEGRALS OFKAHAN-HIROTA-KIMURA DISCRETIZATIONS
REN´E ZANDERInstitut f¨ur Mathematik, MA 7-2Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany
Abstract.
We consider an Ansatz for the study of the existence of formal integrals ofmotion for Kahan-Hirota-Kimura discretizations. In this context, we give a combinatorialproof of the formula of Celledoni-McLachlan-Owren-Quispel for an integral of motion ofthe discretization in the case of cubic Hamiltonian systems on symplectic vector spaces andPoisson vector spaces with constant Poisson structure. Introduction
The purpose of this paper is to explore the combinatorial structure that ensures theexistence of a (formal) modified integral for the Kahan-Hirota-Kimura discrectization of cubicHamiltonian systems on symplectic vector spaces and Poisson vector spaces with constantPoisson structure.The Kahan-Hirota-Kimura discretization scheme has been introduced independently byKahan [9], who applied this method to the Lotka-Volterra system, and Hirota, Kimura,who applied it to the Euler top [7] and the Lagrange top [8]. While the integrability inthe Lotka-Volterra case is still an open question, the Kahan-Hirota-Kimura discretizationscheme produces integrable maps for the Euler top and the Lagrange top. Petrera, Pfadler,Suris deepened the investigation of the integrability of Kahan-Hirota-Kimura discretizationsand provided an extensive list of results for concrete algebraically completely integrablesystems [11–13].The Kahan-Hirota-Kimura discretization scheme can be applied to any system of ordinarydifferential equations ˙ x = f ( x ) for x : R → R n with f ( x ) = Q ( x ) + Bx + c, x ∈ R n . (1)Here each component of Q : R n → R n is a quadratic form, while B ∈ R n × n and c ∈ R n .Then the Kahan-Hirota-Kimura discretization is given by e x − x ǫ = Q ( x, e x ) + 12 B ( x + e x ) + c, (2)where Q ( x, e x ) = 12 ( Q ( x + e x ) − Q ( x ) − Q ( e x )) , is the symmetric bilinear form corresponding to the quadratic form Q . Here and below weuse the following notational convention which will allow us to omit a lot of indices: for asequence x : Z → R we write x for x k and e x for x k +1 . Equation (2) is linear with respect E-mail: [email protected] to e x and therefore defines a rational map e x = Φ( x, ǫ ). Clearly, this map approximates thetime-(2 ǫ )-shift along the solutions of the original differential system. (We have chosen aslightly unusual notation 2 ǫ for the time step, in order to avoid appearance of powers of 2in numerous formulae; a more standard choice would lead to changing ǫ ǫ/ x ↔ e x with the simultaneoussign inversion ǫ
7→ − ǫ , one has the reversibility property Φ − ( x, ǫ ) = Φ( x, − ǫ ) . In particular,the map Φ is birational. As already known to Kahan, the explicit form of the map Φ definedby (2) is e x = Φ( x, ǫ ) = x + 2 ǫ ( I − ǫf ′ ( x )) − f ( x ) , (3)where f ′ ( x ) denotes the Jacobi matrix of f ( x ).The study of the integrability of Kahan-Hirota-Kimura discretizations has been continuedby Celledoni, McLachlan, Owren, Quispel [2]. They obtained the following remarkable resultexplaining some of the cases presented in [13]. Let H : R n → R be a cubic Hamiltonian, J ∈ R n × n be a constant skew-symmetric matrix and f ( x ) = J ∇ H ( x ). Then the map (3)possesses the following rational integral of motion: e H ( x, ǫ ) = H ( x ) + 2 ǫ ∇ H ( x )) T ( I − ǫf ′ ( x )) − f ( x ) . (4)In the field of analysis of numerical integrators there is a rich history of the use of formalseries (see [1, 4–6]). In this paper, we consider an Ansatz, proposed by Petrera and Suris,for the study of the existence of integrals of motion for the Kahan-Hirota-Kimura map (2). Definition 1.1. A formal integral for the Kahan-Hirota-Kimura map (3) is a formalpower series e H ( x, ǫ ) = X q ≥ H q ( x ) ǫ q , (5) with smooth functions H q : R n → R , q ∈ N , satisfying the partial differential equations H (1) q − [ f ] = − q − X k =0 q − k X i =1 X j + ··· + j i = q − k ≤ j ≤···≤ j i i − µ ( j , . . . , j i ) H ( i ) k [ f j , . . . , f j i ] , (6) where µ ( j , . . . , j i ) = µ ! µ ! · · · and the integers µ , µ , . . . count equal terms among j , . . . , j i ,i.e. µ , µ , . . . are the multiplicities of the distinct elements k , k , . . . ∈ { j , . . . , j i } in thetuple ( j , . . . , j i ) . Indeed, the Kahan-Hirota-Kimura map (3) has the formal series e x = x + 2 ∞ X n =1 ǫ n f n ( x ) , (7) where f n ( x ) = ( f ′ ( x )) n − f ( x ). Then substituting (7) into (5) and writing the Taylor serieswe obtain e H ( e x, ǫ ) = e H ( x, ǫ ) + X i ≥ i i ! e H ( i ) [ ǫf + ǫ f + ǫ f + · · · ] i = e H ( x, ǫ ) + X i ≥ X j ,...,j i ≥ i i ! ǫ j + ··· + j i e H ( i ) [ f j , . . . , f j i ]= e H ( x, ǫ ) + X k ≥ X i ≥ X j ,...,j i ≥ i i ! ǫ k + j + ··· + j i H ( i ) k [ f j , . . . , f j i ]= e H ( x, ǫ ) + X q ≥ q − X k =0 q − k X i =1 X j + ··· + j i = q − kj ,...,j i ≥ i i ! H ( i ) k [ f j , . . . , f j i ] ǫ q = e H ( x, ǫ ) + X q ≥ q − X k =0 q − k X i =1 X j + ··· + j i = q − k ≤ j ≤···≤ j i i µ ( j , . . . , j i ) H ( i ) k [ f j , . . . , f j i ] ǫ q , Now, for e H being a (formal) integral of motion of the Kahan-Hirota-Kimura map meansthat all coefficients of ǫ q vanish for all q ∈ N . Remark 1.
We put emphasis on the fact that for a formal integral e H one still has to checkits convergence in order to decide whether it is indeed an integral of motion. Remark 2.
At each step q ∈ N the right hand side of (6) depends only on H , . . . , H q − .Hence, the equations (6) can be solved recursively to obtain a formal integral. Example 1.2.
We consider the equations (6) at order ǫ, ǫ and ǫ . ǫ : The equation reads H (1)0 [ f ] = 0 . (8)Thus, H is an integral of the continuous system. ǫ : The equation reads H (1)1 [ f ] = − (cid:16) H (1)0 [ f ] + H (2)0 [ f , f ] (cid:17) = − ( H (1)0 [ f ]) (1) [ f ] = 0 . (9)Thus, we may choose H = 0. ǫ : The equation reads H (1)2 [ f ] = − (cid:16) H (1)1 [ f ] + H (2)1 [ f , f ]+ H (1)0 [ f ] + 2 H (2)0 [ f , f ] + 23 H (3)0 [ f , f , f ] (cid:19) . Using condition (9) we get0 = ( H (1)1 [ f ]) (1) [ f ] = H (1)1 [ f ] + H (2)1 [ f , f ] = 0 REN´E ZANDER and obtain H (1)2 [ f ] = − (cid:18) H (1)0 [ f ] + 2 H (2)0 [ f , f ] + 23 H (3)0 [ f , f , f ] (cid:19) . (10)Now, we study the existence of solutions H to equation (10). Applying condition(8) we get 0 = ( H (1)0 [ f ]) (1) [ f ] = H (1)0 [ f ] + H (1)0 [ f , f ]and equation (10) can be written as H (1)2 [ f ] = − (cid:18)
23 ( H (2)0 [ f , f ]) (1) [ f ] + 13 H (1)0 [ f ] (cid:19) . This means that there exists a solution H if there is a function η such that η (1) [ f ] = H (1)0 [ f ]. In the case of a Hamiltonian system on a symplectic vector space or Poissonvector space with constant Poisson structure (i.e., f = J ∇ H ) by the skew-symmetryof J we get H (1)0 [ f ] = ∇ H T J H ′′ J H ′′ J ∇ H = 0 . Thus, in this case we can assign H = − H (2)0 [ f , f ].The purpose of the study of equations (6) is to further the understanding of the mechanismthat ensures (or prevents) that the Kahan-Hirota-Kimura discretization admits integrals ofmotion. While experiments indicate that for the study of obstructions for the existence offormal integrals it suffices to consider (6) at low orders, to prove the existence of formalintegrals one has to check that equations (6) are satisfied for all q ∈ N . A natural startingpoint for this investigation is given by the following claim. Proposition 1.3.
Let H : R n → R be a cubic Hamiltonian, J ∈ R n × n be a constant skew-symmetric matrix and f = J ∇ H . Define the smooth functions H l : R n → R by H k = 23 ( − k H (2)0 [ f k , f k ] and H k − = 0 , for k ∈ N . (11) Then, for all q ∈ N , we have q − X k =0 q − k X i =0 X j + ··· + j i = q − k i µ ( j , . . . , j i ) H ( i ) k [ f j , . . . , f j i ] = 0 , (12) where µ ( j , . . . , j i ) = µ ! µ ! · · · and µ , µ , . . . count equal terms among j , . . . , j i . Although the above statement is a consequence of (4), the combinatorial structure thatensures the solvability of the partial differential equations (12) remains rather mysterious.In this paper, we develop a combinatorial proof of this statement based on the formalism oftrees described by Hairer, Lubich, Wanner [1]. Note that the application of this formalism istied to the fact already noted in [2] that the solution can be expressed in terms of elementaryHamiltonians. Trees
In this part we recall the notion of trees (related to B-series) following [1,5,6]. We considerordinary differential equations ˙ x = f ( x ) with Hamiltonian vector field f = J ∇ H for cubicHamiltonians H : R n → R and constant skew-symmetric J ∈ R n × n . Definition 2.1.
The set T = { , , , , , , , , . . . } of rooted (unordered) trees is recursively defined by ∈ T, [ τ , . . . , τ m ] ∈ T, for all τ , . . . , τ m ∈ T, where is the tree with only one vertex, and τ = [ τ , . . . , τ m ] represents the tree obtained bygrafting the roots of τ , . . . , τ m by additional edges to a new vertex which becomes the root of τ . The order | τ | of a tree τ is its number of vertices. A collection F of rooted trees is called forest . Remark 3.
Note that τ = [ τ , . . . , τ m ] does not depend on the ordering of τ , . . . , τ n , forexample, [ , [ ]] = and [[ ] , ] = are equal in T .We use the notation V( τ ) for the set of all vertices and E( τ ) for the set of all edges of τ ∈ T . We write e = ( ν, ν ′ ) for the edge linking ν and ν ′ . Given τ ∈ T we write r ( τ ) ∈ V( τ )for the root of τ . By deg( ν ) we denote the number of edges attached to ν ∈ V( τ ). Definition 2.2. An isomorphism of trees τ , τ is a map φ : V( τ ) → V( τ ) such that ( ν, ν ′ ) ∈ E( τ ) if and only if ( φ ( ν ) , φ ( ν ′ )) ∈ E( τ ) . Trees τ , τ are called properly isomor-phic (we write τ = τ ) and there is an isomporhism φ : τ → τ with φ ( r ( τ )) = r ( τ ) . The set T can be seen as set of equivalence classes of properly isomorphic trees. Definition 2.3.
For a tree τ = [ τ , . . . , τ m ] ∈ T the symmetry coefficient σ ( τ ) is definedrecursively by σ ( ) = 1 , σ ( τ ) = σ ( τ ) · · · σ ( τ m ) µ ! µ ! · · · , where the integers µ , µ , . . . count equal trees among τ , . . . , τ m . For a forest F the symmetrycoefficient is defined by σ ( F ) = Y τ ∈F σ ( τ ) . Definition 2.4.
For a tree τ ∈ T the branching number b( τ ) is defined by b( τ ) =deg( r ( τ )) and the branching factor α ( τ ) is defined by α ( τ ) = 2 b( τ ) . Definition 2.5.
Let u, v ∈ T , with u = [ u , . . . , u m ] , v = [ v , . . . , v n ] and γ = ( ν , . . . , ν n ) ∈ V( u ) n . The merging product u × γ [ v , . . . , v n ] is given by the tree obtained from u , wherethe rooted subtrees v , . . . , v n of v are attached by a new edge to the vertices ν , . . . , ν n of u respectively. By abuse of notation we write u × γ v meaning that a representation v =[ v , . . . , v n ] is fixed. The Butcher product is defined as u ◦ v = [ u , . . . , u m , v ] . Definition 2.6.
For a given smooth vector field f : D → R n (with open D ⊂ R n ) and for τ ∈ T we define the elementary differential F ( τ ) : D → R n by F ( )( x ) = f ( x ) , F ( τ )( x ) = f ( m ) ( x ) ( F ( τ )( x ) , . . . , F ( τ m )( x )) for τ = [ τ , . . . , τ m ] . REN´E ZANDER
Definition 2.7.
For a given smooth function H : D → R (with open D ⊂ R n ) and for τ ∈ T we define the elementary Hamiltonian H ( τ ) : D → R by H ( )( x ) = H ( x ) , H ( τ )( x ) = H ( m ) ( x ) ( F ( τ )( x ) , . . . , F ( τ m )( x )) for τ = [ τ , . . . , τ m ] . Here, F ( τ i )( x ) are elementary differentials corresponding to f ( x ) = J ∇ H ( x ) . Note that the solution given by (11) can be given in terms of elementary Hamiltonians.The following lemma provides relations among elementary Hamiltonians that are essentialfor the validity of the relations (12). We may stress the fact that we are in the situationof Hamiltonian systems on symplectic vector spaces or Poisson vector spaces with constantPoisson structure.
Lemma 2.8. [1] Elementary Hamiltonians satisfy H ( u ◦ v )( x ) + H ( v ◦ u )( x ) = 0 for all u, v ∈ T. (13) In particular, we have H ( u ◦ u )( x ) = 0 for all u ∈ T .Proof. Let u = [ u , . . . , u m ] ∈ T and v = [ v , . . . , v n ] ∈ T . Then using the skew-symmetry of J we have H ( u ◦ v ) = H ( m +1) ( F ( u ) , . . . , F ( u m ) , F ( v ))= F ( v ) T ( ∇ H ) ( m ) ( F ( u ) , . . . , F ( u m ))= − F ( u ) T ( ∇ H ) ( n ) ( F ( v ) , . . . , F ( v n ))= − H ( n +1) ( F ( v ) , . . . , F ( v m ) , F ( u )) = − H ( v ◦ u ) . (cid:3) Remark 4.
As a consequence of this lemma we note that for trees u, v ∈ T which have thesame graph and differ only in the position of the root, we have H ( u )( x ) = ( − κ ( u,v ) H ( v )( x ),where κ ( u, v ) is the number of (1 step) root changes that are necessary to obtain u from v . Definition 2.9.
Trees τ , τ ∈ T are called equivalent (we write τ ∼ = τ ) if they have thesame graph and differ only in the position of the root (i.e. τ , τ are isomorphic but notproperly isomorhic). We denote the set of equivalence classes by T . Definition 2.10.
Let H : D → R (with open D ⊂ R n ) be a smooth function; τ = [ τ . . . , τ m ] and t = [ t , . . . , t n ] be trees in T . Then we define the derivative of H ( τ ) w.r.t t as H ( τ )[ t ] : D → R by H ( τ )[ ]( x ) = H ( τ )( x ) , H ( τ )[ t ]( x ) = ( H ( τ )) ( n ) ( F ( t )( x ) , . . . , F ( t n )( x )) . Lemma 2.11.
Let H : D → R (with open D ⊂ R n ) be a smooth function; τ = [ τ . . . , τ m ] and t = [ t , . . . , t n ] be trees in T . Then we have H ( τ )[ t ]( x ) = X γ =( ν ,...,ν n ) ∈ V ( τ ) n H ( τ × γ t )( x ) . (14) Proof.
This is a consequence of Leibniz’ rule for derivatives. (cid:3) Proof of proposition 1.3
In this part, we present a new combinatorial proof of proposition 1.3. First, we give areformulation of this statement using the formalism of trees. To do this in a convenient wayit is necessary to introduce some further notation.
Definition 3.1.
Let T ′ = { τ ∈ T : deg( ν ) ≤ for all ν ∈ V( τ ) and b( τ ) = 1 } be the set of tall trees , i.e., T ′ = { , , , , . . . } . Definition 3.2.
Let T ′′ = { τ = [ τ , . . . , τ m ] ∈ T : τ i ∈ T ′ , for ≤ i ≤ m } be the set of treeswith branching only at the root, i.e., T ′′ = { , , , , , , , , . . . } . We write θ = , θ k = [ τ k , τ k ], where τ k is the tall tree with k vertices, and θ k − = ∅ for k ∈ N . For example: θ = , θ = , θ = , . . . We write T k for the subset of trees with k vertices of the set T , and similarly for T ′′ k .Now, we consider the following reformulation of proposition 1.3. Proposition 3.3.
Let H : R n → R be cubic Hamiltonian, J ∈ R n × n be a constant skew-symmetric matrix and f = J ∇ H . Define the smooth functions H l : R n → R by H k = 23 ( − k H ( θ k ) and H k − = 0 , for k ∈ N . (15) Then, for all m ∈ N , we have m − X k =0 X t ∈ T ′′ m +1 − k α ( t ) σ ( t ) H k ( t ) = 0 . (16) Example 3.4.
We consider the case m = 3. Then equation (16) reads0 = 23 H ( ) + 2 H ( ) + H ( ) + H ( ) + H ( ) + H ( )Now, we substitute H and H by the functions given by (15). Then we obtain0 = 23 H ( ) + 2 H ( ) + H ( ) − H ( )[ ]= 23 H ( ) + H ( ) . This is an identity since the trees in the last line are equivalent to ◦ . REN´E ZANDER
Remark 5.
Taking into account lemma 2.11 we immediately see that for each m ∈ N equation (16) reads X τ ∈ T m +1 a ( τ ) H ( τ ) = 0 , (17)with rational coefficients a ( τ ) ∈ Q . Now, using remark 4 we can write (17) as X ¯ τ ∈ T m +1 X ˜ τ ∈ ¯ τ a (˜ τ )( − κ (˜ τ,τ ) ! H ( τ ) = 0 . (18)Now, at each step m ∈ N all summands for ¯ τ ∈ T in equation (18) vanish. This can bechecked using the following observations.(1) In the situation of a cubic Hamiltonian H : R n → R and a quadratic vector field f = J ∇ H we have that H ( τ ) = 0 if there is a vertex in τ ∈ T with degree morethan 3.(2) H ( τ ) = 0 if τ is equivalent to u ◦ u for any tree u ∈ T (lemma 2.8).(3) H ( τ ◦ τ ) = − H ( τ ◦ τ ) for τ , τ ∈ T (lemma 2.8), i.e., the values of elementaryHamiltonians of equivalent trees differ only in the sign.The first two observations will lead to the following definition of admissible trees. The lastobservation will require the development of a good way to enumerate all trees (and thecorresponding coefficients in equation (16) in an equivalence class of admissible trees. Thiswill require the introduction of some notation in the following. Definition 3.5.
A tree τ ∈ T is admissible if deg( ν ) ≤ for all ν ∈ V( τ ) and τ is notequivalent to τ ′ ◦ τ ′ for any τ ′ ∈ T . We will denote the set of admissible trees by T ∗ ⊂ T . Definition 3.6.
Given a tree τ ∈ T ∗ a labeling is a map ℓ : V( τ ) → N such that ( − ℓ ( ν ) = − ( − ℓ ( ν ′ ) for adjacent vertices ν, ν ′ ∈ V( τ ) . Remark 6.
This definition of labeling is necessary to keep track of the different signs ofthe elementary Hamiltonians corresponding to the trees in an equivalence class of admissibletrees.
Definition 3.7.
Given τ ∈ T ∗ and a subtree w ⊂ τ by τ w we denote the tree obtained bycontracting all vertices of w to one vertex which becomes the root of τ w . We define the setof proper subtrees of the tree τ by W ( τ ) = { w ⊂ τ : τ w ∈ T ′′ and w ≡ θ k , for some k ∈ N } . We use the notation w ⊂ τ for a strict subset w of τ . The trees w ∈ W ( τ ) are viewedas rooted trees w ∈ T with the middle vertex as root. For w ∈ W ( τ ) , we write τ w ∈ ¯ τ for the tree in the equivalence class ¯ τ of τ that has the middle node of w as root. Subtrees w, w ′ ∈ W ( τ ) are called equivalent if and only if there is an automorphism of τ that restrictsto an isomorphism of w and w ′ . The set of equivalence classes is denoted by W ( τ ) . We use the notation W ′ ( τ ) = { w ∈ W ( τ ) : | w | > } and similarly for W ′ ( τ ). Remark 7.
Let τ ∈ T ∗ . Then it is easy to see that for equivalent subtrees w, w ′ ∈ W ( τ )we have τ w = τ w ′ and τ w = τ w ′ . So, equivalent subtrees correspond to the same e τ ∈ ¯ τ .Moreover, given any labeling ℓ : V( τ ) → N we have that ( − ℓ ( r ( w )) = ( − ℓ ( r ( w ′ )) . For thelast observation it is important that we exclude trees τ equivalent to τ ′ ◦ τ ′ for any τ ′ ∈ T . Definition 3.8.
Given τ ∈ T ∗ and a subtree w ∈ W ( τ ) we count the number of waysobtaining τ w as merging product w × γ τ w with ω τ ( w ) = |{ γ ∈ V( w ) b( τ w ) : w × γ τ w = τ w }| . We note that ω τ ( w ) does not depend on the choice of a representative w ∈ ¯ w . Example 3.9.
We consider an admissible tree τ ∈ T ∗ and all subtrees w , w , w ∈ W ′ ( τ ). (a) w ⊂ τ (b) w ⊂ τ (c) w ⊂ τ We notice that w and w are equivalent. For w (and w ) we have w =
21 3 , τ w = , τ w = . To illustrate the computation of ω τ ( w ) we have labeled the vertices of w . Then γ = (1 , γ = (2 , γ = (2 , γ = (3 ,
2) are all tuples of vertices such that w × γ τ w = τ w , i.e., ω τ ( w ) = 4.For w we have w =
21 3 , τ w = , τ w = . Again, we have labeled the vertices of w . Then γ = (1 , , γ = (3 ,
3) are all tuples ofvertices such that w × γ τ w = τ w , i.e., ω τ ( w ) = 2. Definition 3.10.
Given τ ∈ T ∗ and a subtree w ∈ W ( τ ) we define c ( w ) = if | w | = 1 , ( − | w |− if | w | > . Lemma 3.11.
The following identity holds m − X k =0 X t ∈ T ′′ m +1 − k α ( t ) σ ( t ) H k ( t ) = X ¯ τ ∈ T ∗ m +1 X ¯ w ∈ W α ( τ w ) σ ( τ w ) c ( w )( − ℓ ( r ( w )) ω τ ( w ) H ( τ ) . (19) Proof.
We write (16) in a different way. Let c = 1 , c k = 23 ( − k , c k − = 0 , for k ∈ N . Then using lemma 2.11 we have m − X k =0 X t ∈ T ′′ m +1 − k α ( t ) σ ( t ) H k ( t ) = m − X k =0 X t ∈ T ′′ m +1 − k α ( t ) σ ( t ) c k X γ ∈ V( θ k ) b( t ) H ( θ k × γ t ) = X ¯ τ ∈ T ∗ m +1 m − X k =0 X t ∈ T ′′ m +1 − k α ( t ) σ ( t ) c k Ω t,k ( τ ) H ( τ ) , where Ω t,k ( τ ) = X e τ ∈ ¯ τ ( − κ ( τ, e τ ) |{ γ ∈ V( θ k ) b( t ) : θ k × γ t = e τ }| . Given τ ∈ T ∗ there is a one-to-one correspondence between triples ( t, k, e τ ) and equivalenceclasses ¯ w . This yields the proof. (cid:3) Remark 8.
Lemma 3.11 enhances us with a good way to enumerate and sum up the co-efficients of all trees in an equivalence class of admissible trees. In the next part we showthat C ( τ ) = X ¯ w ∈ W ( τ ) α ( τ w ) σ ( τ w ) c ( w )( − ℓ ( r ( w )) ω τ ( w ) = 0 (20)for all τ ∈ T ∗ . Therefor, we distinguish three cases of admissible trees τ ∈ T ∗ by thestructure of their graphs:(1) Let τ ∈ T ∗ with deg( ν ) ≤ ν ∈ V( τ ).(2) Let τ ∈ T ∗ with deg( ν ) = 3 for exactly one vertex ν ∈ V( τ ).(3) Let τ ∈ T ∗ with deg( ν ) = 3 for at least two vertices ν ∈ V( τ ).Then we prove equation (20) for each case. This finishes the proof of proposition 3.3.4. Proof of equation (20)
The following technical lemma will be essential for the proof of equation (20) in all threecases.
Lemma 4.1.
Let m, n ∈ N and S emn = { ( i, j ) ∈ Z : 0 ≤ i ≤ m, ≤ j ≤ n, i + j even } and S omn = { ( i, j ) ∈ Z : 0 ≤ i ≤ m, ≤ j ≤ n, i + j odd } . Then we have X ( i,j ) ∈ S ( − j − ( δ i + δ im + δ j + δ jn ) = 0 , S = S emn or S = S omn . (21) Proof.
The claim is true for the latices S e , and S o , (figure (1)). Then by gluing togetherlattices of type S e , or S o , respectively we see that the claim holds for all lattices S with m, n even. Further we see that the claim is true for the lattices S e , and S o , . Now, by asuitable gluing procedure we obtain that the claim holds for all lattices S with m, n ∈ N . (cid:3)
14 14 −
14 14 12 − − Figure 1. S e , and S o , Corollary 4.2.
Let m ∈ N and S m = { i ∈ Z : 0 ≤ i ≤ m } . Then we have X i ∈ S m ( − i − ( δ i + δ im ) = 0 . (22) Proof.
This follows from lemma 4.1 by contracting the lattice S em, to S m . (cid:3) We distinguish three types of graphs of admissible trees τ ∈ T ∗ .(1) Let τ ∈ T ∗ with deg( ν ) ≤ ν ∈ V( τ ).In this situation the cardinality | τ | is always odd (otherwise τ ∼ = τ ′ ◦ τ ′ for some τ ′ ∈ T ). The graph of τ is illustrated in figure (2). Note that we label the verticessequently by ν − a , . . . , ν a . Let w ⊂ τ be the subtree consisting of the vertex ν . ν − ν ν − a ν a ν w Figure 2.
Type (1) graph.(2) Let τ ∈ T ∗ with deg( ν ) = 3 for exactly one vertex ν ∈ V( τ ).The graph of τ is illustrated in figure (3). Let ν denote the unique vertex withdegree equal to 3. (Then the tree τ ∈ T ∗ corresponds to the choice of ν as root).Let w ⊂ τ be the subtree consisting only of the vertex ν . We denote the branchesattached to ν by α , α , α . We write a i = | α i | , for i = 1 , , α α α ν w Figure 3.
Type (2) graph.(3) Let τ ∈ T ∗ with deg( ν ) = 3 for at least two vertices ν ∈ V( τ ).The graph of τ is illustrated in figure (4). Let ν, ν be the extremal vertices withdegree equal to 3, i.e., all other vertices with degree equal to three lie on the segmentbetween ν and ν . Let w denote the subtree connecting ν and ν . (The tree τ ∈ T ∗ corresponds to the choice of a vertex ν ⊂ w as root). We denote the branchesattached to ν and ν by α , α and α , α respectively. We write a i = | α i | , for i =1 , . . . , ν ν να α α α w Figure 4.
Type (3) graph.
Definition 4.3.
Let τ ∈ T ∗ and w ∈ W ( τ ) , with | w | > . Then the pair ( τ, w ) is called symmetric if there is an automorphism of φ ∈ Aut( τ ) that restricts to an automorphism φ | w ∈ Aut( w ) of w ⊂ τ such that φ | w = r , where r ∈ Aut( w ) denotes the reflection at themiddle vertex of w . We define the symmetry coefficient of the pair ( τ, w ) by ς ( τ, w ) = if ( τ, w ) symmetric , if ( τ, w ) non-symmetric . Lemma 4.4.
Let τ ∈ T ∗ and w ∈ W ( τ ) , with | w | > . Then we have ω τ ( w ) = ς ( τ, w ) σ ( τ w ) σ ( τ \ E( w )) , (i) | w | = ς ( τ, w )2 | Aut( τ ) | σ ( τ \ E( w )) , (ii) where τ \ E( w ) denotes the forest obtained from τ by deleting all edges contained in w ⊂ τ .Proof. We prove both claims.(i) Let τ w = [ τ , . . . , τ N ].Observe that if τ n = τ n ′ , for 1 ≤ n, n ′ ≤ N , and γ = ( . . . , ν n , . . . , ν n ′ , . . . ) and γ ′ =( . . . , ν n ′ , . . . , ν n , . . . ) is the same tuple of vertices with ν n and ν n ′ interchanged, then w × γ τ w = w × γ ′ τ w . This yields the factor σ ( τ w ).Suppose that τ n = τ n ′ , for 1 ≤ n, n ′ ≤ N , and ν n = ν n ′ . Then the tuples γ =( . . . , ν n , . . . , ν n ′ , . . . ) and γ ′ = ( . . . , ν n ′ , . . . , ν n , . . . ) are actually identical. Hence, we divideby σ ( τ \ E( w )).Suppose that the vertices of w are sequently labeled by 1 , . . . , | w | . For γ ∈ V( w ) wedenote by ¯ γ the tuple of vertices obtained from γ by replacing each ν ∈ γ with | w | + 1 − ν .Obviously, we have w × γ τ w = w × ¯ γ τ w . So, this yields the factor 2 in the non-symmetriccase (i.e. τ is non-symmetric or τ is symmetric and ( τ, w ) is non-symmetric). If ( τ, w ) issymmetric ¯ γ is already counted with σ ( τ w ).(ii) We divide the total number of automorphisms of τ by the number of automorphismsthat restrict to an automorphism of w ⊂ τ . Then the claim follows by the orbit-stabilizertheorem. (cid:3) Corollary 4.5.
Let τ ∈ T ∗ and w ∈ W ( τ ) , with | w | > . Then we have ω τ ( w ) | Aut( τ ) | = 2 σ ( τ w ) | w | . (23) Proof.
This is a direct consequence of lemma 4.4. (cid:3)
Now, we are in the position to complete the proof of equation (20).
Proof.
We proof all claims.(1) Applying corollary 4.5 and taking into account that for | w | = 1 we have | w | = 2 /σ ( τ w )and ω τ ( w ) = 1 we obtain C ( τ ) = 12 X w ∈ W ( τ ) | w | =1 α ( τ w ) c ( w )( − ℓ ( r ( w )) + X w ∈ W ( τ ) | w | > α ( τ w ) c ( w )( − ℓ ( r ( w )) . Let w ij ∈ W ( τ ) denote the subtree connecting the vertices ν i and ν j , for i ≤ j . Then, for w = w ij , we have: (i) By definition of c ( w ) and α ( τ w ) we compute c ( w ) α ( τ w ) = α ( τ w )2 − ( δ − ai + δ ai ) if | w | = 1 , α ( τ w )( − j − i − ( δ − ai + δ aj ) if | w | > . Note that if w ends at ν − a (i.e. i = − a ) the branching number b( τ w ) is reduced by1 compared to b( τ w ). The same holds if w ends at ν a (i.e., j = a ).(ii) We can assume that ( − ℓ ( r ( w )) = ( − i + j .Finally, we have X w ∈ W ( τ ) | w | =1 α ( τ w ) c ( w )( − ℓ ( r ( w )) = α ( τ w ) X w ∈ W ( τ ) | w | =1 ( − i − ( δ − ai + δ ai ) , (24)and X w ∈ W ( τ ) | w | > α ( τ w ) c ( w )( − ℓ ( r ( w )) = 2 α ( τ w )3 X w ∈ W ( τ ) | w | > ( − j − ( δ − ai + δ aj ) . (25)Using corollary 4.2 we see that (24) and (25) vanish. Note that in (25) the sum vanishesalong every diagonal with j − i = c , for c ∈ N .(2) Applying corollary 4.5 and taking into account that | w | = ω τ ( w ) = 1 we obtain C ( τ ) = α ( τ w ) c ( w )( − ℓ ( r ( w )) | Aut( τ ) | + 2 | Aut( τ ) | X w ∈ W ′ ( τ ) α ( τ w ) c ( w )( − ℓ ( r ( w )) . Let w klij ∈ W ( τ ) denote the subtree connecting the i th vertex of the branch α k and the j th vertex of the branch α l , for k, l = 1 , ,
3, and W kl = { w klij ∈ W ( τ ) } . Then, for w = w klij ,we have:(i) By definition of c ( w ) and α ( τ w ) we compute c ( w ) α ( τ w ) = α ( τ w ) if | w | = 1 , α ( τ w )( − i + j − ( δ iak + δ jal ) if | w | > . Note that if w ends at the a k th vertex of α k (i.e., i = a k ) the branching number b( τ w )is reduced by 1 compared to b( τ w ). The same holds if w ends at a l th vertex of α l (i.e., j = a l ).(ii) We can assume that ( − ℓ ( r ( w )) = ( − j − i .(iii) Passing from summation over all w ∈ W ′ ( τ ) to summation over all w ∈ W ′ kl , for1 ≤ k < l ≤
3, we divide by 2 δ i + δ j since the affected subtrees (with i = 0 or j = 0)are counted multiple.Finally, we obtain C ( τ ) = 4 α ( τ w )3 | Aut( τ ) | X ≤ k 14 + X w ∈ W ′ kl ( − j − ( δ iak + δ jal + δ i + δ j ) . Now, the claim follows by lemma 4.1. (3) Applying corollary 4.5 we obtain C ( τ ) = 2 | Aut( τ ) | X w ∈ W ( τ ) α ( τ w ) c ( w )( − ℓ ( r ( w )) . Let w klij ∈ W ( τ ) denote the subtree connecting the i th vertex of the branch α k and the j th vertex of the branch α l , for k = 1 , l = 3 , 4, and W kl = { w klij ∈ W ( τ ) } . Then, for w = w klij , we have:(i) By definition of c ( w ) and α ( τ w ) we compute c ( w ) α ( τ w ) = 23 α ( τ w )( − i + j + | w |− − ( δ iak + δ jal ) . Note that if w ends at the a k th vertex of α k (i.e., i = a k ) the branching number b( τ w )is reduced by 1 compared to b( τ w ). The same holds if w ends at a l th vertex of α l (i.e., j = a l ).(ii) We can assume that ( − ℓ ( r ( w )) = ( − j − i −| w | +12 .(iii) Passing from summation over all w ∈ W ( τ ) to summation over all w ∈ W kl , for k = 1 , l = 3 , 4, we divide by 2 δ i + δ j since the affected subtrees (with i = 0 or j = 0) are counted multiple.Finally, we obtain C ( τ ) = 3 α ( τ w )4 | Aut( τ ) | X k =1 , , l =3 , X w ∈ W kl ( − j − ( δ iak + δ jal + δ i + δ j ) . Again, the claim follows by lemma 4.1. This finishes the proof. (cid:3) Remark 9. Note that while in the situation of case (1) and (3) the factor 2 / H k , for k ∈ N , is not needed, in the situation of case (2) it is essential forthe vanishing of C ( τ ). Example 4.6. We illustrate equation (20) in table (1–3). w α ( τ w ) σ ( τ w ) c ( w ) ω τ ( w ) ( − ℓ ( r ( w )) − 14 2 1 1 14 2 − − − Table 1. Case (1): ¯ w ∈ W ( τ ) and coefficients. w α ( τ w ) σ ( τ w ) c ( w ) ω τ ( w ) ( − ℓ ( r ( w )) − − − − Table 2. Case (2): ¯ w ∈ W ( τ ) and coefficients. w α ( τ w ) σ ( τ w ) c ( w ) ω τ ( w ) ( − ℓ ( r ( w )) 16 6 − 12 18 6 − Table 3. Case (3): ¯ w ∈ W ( τ ) and coefficients.5. Conclusions In this paper, we have considered an Ansatz for the study of the existence of formalintegrals for Kahan-Hirota-Kimura discretizations. We have used the formalism of trees(related to B-series) to develop a combinatorial proof of formula (4) for a modified integralof the Kahan-Hirota-Kimura discretization in the case of Hamiltonian systems on symplecticvector spaces or Poisson vector spaces with constant Poisson structures. The proof presentedin this work gives insights into the combinatorial structure that ensures the existence of amodified integral in this case.Goals for future research are to deepen the study of the partial differential equations (6)in order to further the understanding of the mechanism that ensures (or prevents) that theKahan-Hirota-Kimura discretization admits integrals of motion. Acknowledgements The author would like to thank Matteo Petrera and Yuri Suris for inspiring discussionsand their critical feedback on this manuscript. References [1] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms forOrdinary Differential Equations , Springer Series in Comput. Mathematics, Vol. 31, Springer-Verlag (2002).[2] E. Celledoni, R.I. McLachlan, B. Owren, G.R.W. Quispel Geometric properties of Kahan’s method , J.Phys. A (2013).[3] E. Celledoni, R.I. McLachlan, D.I. McLaren, B. Owren, G.R.W. Quispel, Integrability properties ofKahan’s method , J. Phys. A (2014).[4] P. Chartier, E. Faou, A. Murua, An algebraic approach to invariant preserving integrators: The case ofquadratic and Hamiltonian invariants A. Numer. Math. (2006), 575-590.[5] P. Chartier, E. Hairer, G. Vilmart, A substitution law for B-series vector fields , INRIA Research ReportNo. 5498 (2005).[6] P. Chartier, E. Hairer, G. Vilmart, Algebraic structures of B-series , Foundations of Computational Math-ematics (2010), 407-427.[7] R. Hirota, K. Kimura, Discretization of the Euler top , J. Phys. Soc. Japan (2000), No. 3, 627–630.[8] K. Kimura, R. Hirota, Discretization of the Lagrange top , J. Phys. Soc. Japan (2000), No. 10, 3193–3199.[9] W. Kahan, Unconventional numerical methods for trajectory calculations , Unpublished lecture notes,(1993).[10] W. Kahan, R.C. Li, Unconventional schemes for a class of ordinary differential equations - with appli-cations to Korteweg-de Vries equation , Jour. Comp. Phys. (1997), 316-331.[11] M. Petrera, Yu.B. Suris, On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top ,Math. Nachr. (2010), No. 11, 1654–1663.[12] M. Petrera, A. Pfadler, Yu.B. Suris, On integrability of Hirota-Kimura-type discretizations: experimentalstudy of the discrete Clebsch system , Exp. Math. (2009), No. 2, 223–247.[13] M. Petrera, A. Pfadler, Yu.B. Suris, On integrability of Hirota-Kimura type discretizations , RegularChaotic Dyn.16