Runge{Kutta methods on Lie groups. Hans Munthe{Kaas 1. We construct generalized Runge{Kutta methods for integration of dierential equations

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1 BIT xx (199x), xxx{xxx. Runge{Kutta methods on Lie groups Hans Munthe{Kaas 1 y 1 Department of Informatics, University of Bergen, Thormhlensgt. 55 N-5020, Norway. Hans.Munthe-Kaasii.uib.no Abstract. We construct generalized Runge{Kutta methods for integration of dierential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolve on the correct manifold. Our methods must satisfy two dierent criteria to achieve a given order: Coecients A i;j and b j must satisfy the classical order conditions. This is done by picking the coecients of any classical RK scheme of the given order. We must construct functions to correct for certain non{commutative eects to the given order. These tasks are completely independent, so once correction functions are found to the given order, we can turn any classical RK scheme into a RK method of the same order on any Lie group. The theory in this paper shows the tight connections between the algebraic structure of the order conditions of RK methods and the algebraic structure of the so called `universal enveloping algebra' of Lie algebras. This may give important insight also into the classical RK theory. 1 Introduction In many areas of applied mathematics one wants to solve dierential equations whose solution is known to evolve on a given manifold. There are two main families of solution techniques for such problems, embedded and intrinsic methods. In the rst of these one embeds the manifold in R n and employs a classical integration scheme here. The problem of this approach is that, except in very special cases, it is generally impossible to nd classical integration schemes which will stay on the correct manifold [5, 9, 11]. The alternative, intrinsic methods, are based on expressing the algorithm via a set of `basic ows' which in each point span all the directions of the manifold. These methods have the advantage of Received August Revised August y This work is sponsored by NFR under contract no /410, through the SYNODE project. WWW:

2 2 Hans Munthe-Kaas being guaranteed to sit on the manifold. The price we must pay for this is that we must be able to compute the basic ows exactly, e.g. by computing matrix exponentials or by integrating `simpler' vector elds dening the basic ows. The topic of this paper is intrinsic integration algorithms in the case where the domain is a Lie group. This has many nice applications discussed in [13, 16]. In [16] we show that if we know how to integrate equations on general Lie groups, we can also integrate equations on virtually any manifold of practical interest (more precisely on any homogeneous manifold). Another application is the construction of explicit orthogonal and unitary integrators. It is known that all classical orthogonal integrators must be implicit [12], while these new algorithms can be explicit. We want to point out some related papers dealing with intrinsic integration algorithms. Iserles [8] presents intrinsic integrators for linear ODEs with variable coecients. This approach can more generally be understood as a numerical implementation of the method of Lie reduction [2, 22], and can be applied to any equation of Lie type. Zanna [21] proposes generalizations of this approach to general ODEs on Lie groups. In [14] we presented a class of generalized Runge{Kutta methods for dierential equations on Lie groups, which we will refer to as MK methods. The main emphasis in [14] was to show how the classical order theory of Butcher could be understood in terms of operations on commutative Lie groups. A related approach is the method of Crouch and Grossman (C-G) [6], who present a general approach and give explicit formulas for a 3. order method. The order theory of their algorithms has recently been systematically developed by Owren and Marthinsen in [17], who also device C-G methods of order 4. The main dierence between the C-G and MK methods is that whereas C-G does the approximations in the Lie group, MK does them in the Lie algebra. I.e. C-G advances by a composed product of exponentials, while MK rst combine elements in the Lie algebra, and then advances by a single exponential mapping applied to this combination. It turns out that the order theory of the MK approach is dramatically simpler than that of the C-G approach. The main reason for this is that whereas operations in the Lie group are nonlinear, the Lie algebra is a linear space. However, the original formulation of MK methods cannot achieve order higher than 2 on a non-commutative Lie group, and it needs some modications. In the present paper we reformulate the theory of [14] in a more precise language, and introduce the necessary modications in order to construct higher order methods for general non-commutative groups. The main idea behind this construction is to devise methods whose order theory is as close as possible to the classical Butcher theory. We obtain a family of methods where two dierent criteria must be satised in order to achieve a given order: Finding coecients A i;j and b j satisfying the classical conditions. This is simply done by picking the coecients of any classical RK scheme of the given order. Finding some correction functions, which correct certain non-commutative eects to the given order.

3 Runge{Kutta methods on Lie groups 3 The two tasks above are completely independent, so once correction functions are found to the given order, we can turn any classical RK scheme into a RK method of the same order on any Lie group. While Marthinsen and Owren [17] prove that for C-G type methods 5 stages are needed to obtain order 4, the present work give examples of 4 stage 4. order algorithms of MK type 1. Readers who want to see the structure of the algorithms without diving too deeply into the mathematics, may jump to Section 3.2, where 3. and 4. order algorithms are explicitly written out. 2 Mathematical background and notation We have chosen to present this paper in the language of general Lie groups rather than e.g. in the more special setting of matrix Lie groups. There are several reasons for this choice. First of all a mathematically rigorous presentation will be an important reference for future work, and it emphasizes that the algorithms are not depending on particular representations. The general language is also precise with respect to the type of the mappings involved, and we avoid the `type errors' which are easily introduced if we employ the language of matrix Lie groups. Thus the abstract language is also an important guide if we want to design object oriented programs for solving dierential equations. The readers who are not familiar with the general notation, may nd it very useful to interpret the formulas for two concrete examples; rst the Abelian case (where our methods reduce to the classical Runge{Kutta methods on R n ) and secondly the matrix Lie group case. All concepts in abstract Lie group theory have a concrete counterpart for matrix Lie groups, which gives a good conceptual handle on the essential ideas. The third important interpretation is the case where the Lie group is a group of dieomorphisms on a manifold and the Lie algebra consists of vector elds tangential to the manifold. The reader will nd the details of these three cases in Section 2.4. The texts [20, 2, 19] are excellent references on Lie group theory. 2.1 Dierential equations on manifolds Let f : N! M be a smooth mapping between manifolds. We let f 0 : TN! T M denote the derivative of f, dened as a mapping between the tangent manifolds. The derivative of a composition of mappings is given as (fg) 0 = f 0 g 0. To avoid a cluttered notation we will identify TR with R itself, thus e.g. if y t : R! M is a curve on M, then y t 0 : R! TM is a curve on TM. A vector eld on M is a smooth mapping F : M! TM, such that F = Id M, where : TM! M is the natural projection. F denes a general initial value problem as: (2.1) y 0 t = F (y t) ; y 0 = p : 1 In a succeeding paper [15], we show how classical RK methods of any order can be modied to a Lie group invariant method of the same order.

4 4 Hans Munthe-Kaas The ow of F is the solution operator F;t : M! M dened in such a way that y t = F;t (p) is the integral curve starting in p, i.e. (2.2) t F;t(p) = F ( F;t (p)) ; F;0 (p) = p : Our goal is to study the numerical integration of (2.1) in the case where M is a Lie group. 2.2 Lie groups and algebras A Lie group is dened as a manifold G equipped with a continuous and smooth group product : GG! G. We dene left and right multiplications in the group as: L a (b) = ab for a; b 2 G R a (b) = ba for a; b 2 G The Lie algebra is dened as the tangent space of G in the identity, g = T Gj e, and has the structure of a (real or complex) linear space equipped with a bilinear skew symmetric form [; ] : gg! g. It is called the Lie bracket, dened below. If G is a matrix Lie group then g is a space of square matrices, and the Lie bracket is the matrix commutator [u; v] = uv? vu. The tangent space at any point a 2 G, T Gj a can be identied with g using either left or right multiplication, i.e. T Gj a = f L 0 a(v) j v 2 g g = f R 0 a(w) j w 2 g g : Thus any vector eld F : G! T G can be written in either of the forms: (2.3) F (a) = L 0 a(f(a)) = R 0 a( ~ f(a)) ; where f; ~ f : G! g. The adjoint representation of G is dened as the mapping 2 Ad : G! Aut(g) given as: (2.4) Ad(a)(v) = (L a R a?1) 0 (v) ; v 2 g: We have f(a) ~ = Ad(a)(f(a)), thus the adjoint mapping is a way of relating left and right representations of vector elds. Note 2.1. All the algorithms of this paper exist in left and right versions. We have based most of the presentation here on the left versions. In some situations it is important to know the right version, and therefore we give the most important formulas and denitions in both forms. The mapping ad : g! End(g) is dened as: (2.5) ad(u) = Ad 0 (e) ; 2 End(g) is the set of all linear maps from g to itself, and Aut(g) those which are invertible.

5 Runge{Kutta methods on Lie groups 5 and the Lie bracket [; ] : gg! g is a bilinear skew symmetric form on g dened as (2.6) [u; v] = ad(u)(v) : Since ad(u) : g! g, we can dene ad n (u) as the n{times iterated map, i.e. ad 0 (u)(v) = v ad n (u)(v) = ad(u)(ad n?1 (u)(v)) = [u; [u; [: : :; [u; v]]]] ; for n The exponential mapping and its dierential The exponential mapping is a mapping exp : g! G. In the case of matrix Lie groups it is given as the matrix exponential (2.12), and on Di(M) it returns the ow of a given vector eld (2.16). The dierential of the exponential turns out to be of major importance in the order theory for Runge{Kutta methods on Lie groups. Since exp : g! G, we must have exp 0 : T g! T G. However, since g is a vector space we can identify T g with g itself, and we have seen that also the tangent space T Gj a can be identied with g. Thus exp 0 can be expressed via a function dexp : g! g. There are two ways of doing this, depending on whether T Gj a is related to g via left of right multiplication. For u; v 2 g we dene: (2.7) (2.8) dexp u (v) = L 0 exp(?u) exp 0 (u; v) d r exp u (v) = R 0 exp(?u) exp 0 (u; v) ; where exp 0 (u; v) = t exp(u + tv) t=0. There are various formulas for expressing dexp u (v) and d r exp u (v): (2.9) (2.10) dexp u = d r exp u = 1X j=0 1X j=0 (?1) j (j + 1)! adj (u) = 1 (j + 1)! adj (u) = Z 1 0 Z 1 0 exp(ad(?su))ds exp(ad(su))ds The sum form of (2.9) is derived in Varadarajan [19] p. 108, while a form similar to the integral form in (2.10) is implicitly derived in the paper of Crouch and Grossman [6] (their Lemma 3). Iserles [8] derives the sum form of (2.10) (his Theorem 1). 2.4 Some important examples of Lie groups Abelian groups and R n Abelian groups are in general groups where ab = ba for all a; b 2 G. This implies that [u; v] = 0 and hence that dexp u (v) = v for all u; v 2 g. Abelian groups are always locally vector spaces, though they may be globally dierent (e.g. tori).

6 6 Hans Munthe-Kaas The real vector spaces R n are special Abelian Lie groups where: (G; ) = (R n ; +) (g; +; [; ]) = (R n ; +; [u; v] = 0) exp(u) = u T G = R n R 0 a(b) = L 0 a(b) = b The initial value problem (2.40) takes the form Matrix Lie groups y 0 t = f(y t ) ; for f : R n! R n. The general linear group, G = GL(n;R), is the group of all real invertible nn matrices, where the product is the matrix product. The Lie algebra g = gl(n; R) consists of all real nn matrices, + in g is the sum of matrices. We have: (2.11) (2.12) (2.13) (2.14) (2.15) [u; v] = uv? vu exp(v) = 1X j=0 v j j! T G = f bv j b 2 G; v 2 g g = f vb j b 2 G; v 2 g g R 0 a(vb) = vba L 0 a(bv) = abv In this case the initial value problem (2.40) takes the form (3.12). Other matrix Lie groups are continuous subgroups of GL(n), and the corresponding algebras are subalgebras of gl(n). Some important cases are: The orthogonal group O(n) = a 2 GL(n;R) j a T a = I. The corresponding algebra consists of all skew symmetric matrices. The special linear group SL(n) = f a 2 GL(n;R) j det a = 1 g, where the algebra consists of all matrices with trace 0. O(n) is important in orthogonal ows and SL(n) is essential for the understanding of volume preserving ows The group Di(M). For a given manifold M, Di(M) consists of all dieomorphisms on M, i.e. smooth invertible mappings : M! M. Its Lie algebra is X(M), i.e. the set of all tangent vector elds on M, and the bracket [; ] is the Lie{Jacobi bracket of vector elds, see [1] for its denition. The product in Di(M) is the composition of dieomorphisms and the sum in X(M) is the pointwise sum of vector elds. The exponential mapping exp : X(M)! Di(M) is given as (2.16) where F;t is given in (2.2). exp(tf ) = F;t ;

7 Runge{Kutta methods on Lie groups The universal enveloping algebra of g The enveloping algebra allow us to introduce higher order dierential operators, and is important in the order theory of RK methods. An element v 2 g denes left and right invariant vector elds on G as: (2.17) (2.18) The ows of X v and Y v are given as: (2.19) (2.20) X v (a) = L 0 a(v) (left invariant) Y v (a) = R 0 a(v) (right invariant) X v ;t Y v;t = R exp(tv) = L exp(tv) Let f be a smooth real valued function on G, f 2 C 1 (G). Then v may act as a derivation of f either from left or right, as the Lie derivative of f with respect to either X v or Y v. I.e. v[]; []v : C 1 (G)! C 1 (G) are dened as: (2.21) (2.22) (v[f])(a) = t f(aexp(tv)) t=0 ([f]v)(a) = t f(exp(tv)a) t=0 We can introduce higher order left and right invariant dierential operators by iterating rst order operators, i.e. we introduce the second order left and right invariant operators 3 (uv)[] = u[v[]] and [](uv) = [[]u]v. It can be shown that (2.23) (2.24) (uv)[f](a) = [f] (uv)(a) = 2 t 1 t 2 f(aexp(t 1 u)exp(t 2 v)) 2 t 1 t 2 f(exp(t 1 u)exp(t 2 v)a) t 1=t 2=0 t 1=t 2=0 The sum of two dierential operators is dened the obvious way, (u + v)[f] = u[f] + v[f], and the 0'th order identity operator Iis dened as I[f] = f. Definition 2.1. The Universal enveloping algebra G of a Lie algebra g consists of all invariant dierential operators generated by fi; gg under the operations and + dened above. For any vector space V we identify T V with V, thus if t : R! V we have t =t = 0 t : R! V. In this case we will henceforth use the notation: (2.25) (i) = i t i t t=0 More specically, for a curve t : R! G, we dene the curve 0 t : R! G such that 0 t[f] = t ( t[f]) for all f 2 C 1 (G). 3 The product uv should not be confused with a matrix product!

8 8 Hans Munthe-Kaas It can be shown that the sum + and the product on G behave in the familiar way under derivations, i.e. ( t + t ) 0 = 0 t + 0 t ( t t ) 0 = 0 t t + t 0 t Note 2.2. We have introduced G in terms of invariant derivations on G. An alternative approach is a purely algebraic construction, see [19]. In this construction, the products uv are tensor products 4. Thus e.g. second order homogeneous elements of G lie in (a quotient space of) gg. The detailed algebraic structure of G is not needed for this paper. However, to understand the tensorial nature of the order conditions for RK methods, it is very useful to look somewhat closer at the enveloping algebra of the s{fold cartesian product of a Lie group. 2.6 Cartesian products of Lie groups Let G s denote the s-fold cartesian product of a Lie group, G s = G G, let g s be its Lie algebra, g s = g g and let G s be the enveloping algebra G s = G G : The elements of G s, g s and G s are s-tuples of G, g and G respectively. All the basic operations such as product on G s, + and [; ] on g s, exp : g s! G s and the product on G s are dened componentwise on the tuples. We have e.g. (2.26)exp(v) = (exp(v 1 ); : : : ; exp(v s )) ; where v = (v 1 ; v 2 ; : : : ; v s ) 2 g s (2.27) u + v = (u 1 + v 1 ; : : :; u s + v s ) ; where u; v 2 g s We introduce the following tensor product basis for g s : g s = R s g ; i.e. g s is the linear span of vectors such as: qv = (q 1 v; q 2 v; : : : ; q s v) ; where q 2 R s ; v 2 g : We will see that this basis can be extended to all of G s. We equip R s with the Schur product, i.e. (2.28) qr = (q 1 r 1 ; q 2 r 2 ; : : : ; q s r s ) T and the identity in R s is given as Obviously, the identity in G s is: 1 = (1; 1; : : :; 1) T : I s = 1I= (I;I; : : :;I) ; 4 Or more precisely a quotient construction on a tensor product space, where elements such as uv? vu? [u; v] are divided out.

9 Runge{Kutta methods on Lie groups 9 and from (2.27) we see that the product in G s is given as (2.29) (qv) (rw) = (qr)(vw) for q; r 2 R s, v; w 2 G. This gives the product on G s in the tensor product basis, and it is extended to all of G s by linearity. For the order theory of Runge{Kutta methods, it is important to note that whereas generally vw 6= wv for v; w 2 G, we always have qr = rq for q; r 2 R s. I.e. the non-commutativity sits only in the right part of the product on G s. 2.7 Equations of Lie type Given a curve t 2 g, an equation (2.30) y t 0 = Xt (y t ) = L 0 y t ( t ) ; y 0 = p ; where the right hand side is a time dependent, left- or right-invariant vector eld, is called an equation of Lie type. Such equations play a central role in the analytical treatment of ODEs. Let s t be the integral curve of (2.30) starting in s 0 = e, where e is the identity in G. This is called the fundamental solution. The general solution of (2.30) is given as y t = ps t. If the right hand side is instead Y t, we obtain the solution y t = s t p. Thus: (2.31) (2.32) X t ;t = R st ; where s 0 t = X t (s t ) ; s 0 = e ; Y t ;t = L st ; where s 0 t = Y t (s t ) ; s 0 = e : Equations of Lie type provide us with a natural 1{1 correspondence between curves on g and curves on G starting in the point p. This will be used in our denition of the order of numerical integration schemes. Lemma 2.1. Given an arbitrary curve y t 2 G, y 0 = p. If t 2 g is given as t = L 0 y?1 (yt) 0 ; t then y t = R st (p) ; where s 0 t = X t (s t ) ; s 0 = e : 2.8 Lie{Butcher series The basic tool for series developments on manifolds is Lie series, which is essentially Taylor series adapted to manifolds. The Lie{Butcher series, introduced in [14], is a special form of Lie series which generalizes the theory of J. Butcher [3, 4] to manifolds. Let F be a vector eld on a manifold M, let F;t be the ow of F and let f be a `geometric object' 5 on M. We now assume that f 2 C 1 (M). Let F;t f 5 It can e.g. be a real function, a tensor eld, a dierential k{form etc.

10 10 Hans Munthe-Kaas denote the pullback of the object along the ow. The denition of the pullback depends on the object. When f 2 C 1 (M) the pullback is simply dened as (2.33) F;t f = f F;t : The Lie derivative of the object f with respect to a vector eld F is generally dened as: (2.34) F [f] = t ( F;t f) t=0 i.e. the rate of change of f at time t = 0, as it pulled back along the ow. By iterating this formula, we nd the basic form of the Lie series: F;t f = f + tf [f] + t2 t3 1X F [F [f]] + 2 3! F [F [F [f]]] + = t i i! F i [f] : i=0 If both the vector eld F and the object f are time dependent, the rate of change at an arbitrary time is given as [1] (p.285) (2.35) t ( F t;t f t) = F t;t ; F t [f t ] + f t t Now, if f is independent of time and F t = X t then (2.35) yields: t 2 Xt ;t f t 2 Xt ;t f = X t ;t ( t [f]) = X t ;t ( t t [f] + 0 t[f]) etc. To express the n{th derivative, we introduce Bt n ( t ) 2 G recursively as: (2.36) B 0 t ( t ) = I (2.37) From (2.35) we see that (2.38) Bt( i t ) = t B i?1 t ( t ) + t n t n X t ;t f? B i?1 t ( t ) ; i > 0 = X t ;t Bn t ( t )[f] : Note that Bt n ( t ) depends only on the derivatives of t up to order n? 1. We dene (2.39) B n (0) ; (1) ; : : : ; (n?1) = B n t n?1 X i=0 t i i! (i)! t=0 B n are n{variate polynomials in G. The rst of these are: B 0 () = I B 1 (0) = (0) B 2 (0) ; (1) = (0) (0) + (1) B 3 (0) ; (1) ; (2) = (0) (0) (0) + 2 (0) (1) + (1) (0) + (2) :

11 Runge{Kutta methods on Lie groups 11 We will use these polynomials and Lemma 2.1 to nd a special type Lie series characterizing the solution y t of the general initial value problem (2.1). Using the left form of (2.3) we can write (2.1) as (2.40) y 0 t = L 0 y t (f(y t )) ; y 0 = p ; where f : G! g. For vector valued functions (such as this f), the denition of pullback and derivations is identical to the denition for functions in C 1 (G) given above 6. Thus from Lemma 2.1 and (2.31), we nd that t 2 g is given as (2.41) t = L 0 y?1 (y 0? t) = f(y t ) = f(ps t ) = R s t f (p) = t Xt ;t f (p) : >From (2.38) we obtain: Theorem 2.2. Let y t be the solution of the initial value problem (2.40) and let t be the corresponding curve on g dened in Lemma 2.1. Then (2.42) t = 1X i=0 t i i! (i) ; where (2.43) (i) = B (i) (0) ; (1) ; : : : ; (i?1) [f](p) and B i are the polynomials dened in (2.39). The series dened in (2.43) is what we call the Lie{Butcher series characterizing the initial value problem (2.40). To see clearer what it means, we introduce the following shorthand notation: For 2 G, dene 2 g as (2.44) = [f](p) : Thus the bar operator is an R{linear map from G to g depending on f and p. Equation (2.43) yields (2.45) (2.46) (2.47) (0) = B 0 () = I (1) = B 1 ( (0) ) = I (2) = B 2 ( (0) ; (1) ) = II+ I (2.48) (3) = III+ 2 II+ II+ II+ I etc. If we turn the bar-patterns upside down, we see the relationship between these operators and the elementary dierentials in the Butcher theory, represented as trees. See Figure 2.1. Note that when the group is non-commutative, then permutation of the branches in the trees yields dierent elementary dierentials. This diers from the classical Butcher theory, where trees with permuted branches are considered identical. 6 Consider f : G! V via its components f i 2 C 1 (G) given relative to some basis for V.

12 12 Hans Munthe-Kaas? I I? I III II?? I? II? II?? II Figure 2.1: Correspondence with the Butcher-tree notation 3 Runge{Kutta methods We will study the numerical solution of (2.40). There exists several ways to generalize classical Runge{Kutta methods to non-commutative Lie groups in such a way that they reduce to the classical methods when we plug in a commutative group. The formulation we study here is deliberately designed such that the order theory becomes as close as possible to the classical order theory. It is close to the formulation in [14], the main dierence being the correction functions i () and () which are new. The choice of correction functions is discussed later. Let s be the number of stages of the method and h the step size and let ^y i be the numerical solution in step i. Our family of generalized RK methods is dened as: Algorithm 3.1. RK-MK : ^y 0 = p for n = 0; 1; : : : for i = 1; 2; : : :; s u i = h P s j=1 A i;jk j ~u i = i (u i ; k 1 ; k 2 ; : : :; k s ) k i = f(^y n exp( ~u i )) end v = h P s j=1 b jk j ~v = (v; k 1 ; k 2 ; : : : ; k s ) ^y n+1 = ^y n exp(~v) end where u i ; ~u i ; k i ; v; ~v 2 g and ^y i 2 G. The real constants A i;j ; b j and the correction functions i () and () determine a particular scheme. The method is explicit if A i;j = 0 for i j and the functions i (u i ; k 1 ; : : :) depend only on k 1 ; k 2 ; : : : ; k i?1. In this case, the iteration over i can be computed explicitly. The classical way of dening the order of RK methods on R n is to say that the method has order q if jj^y 1? y h jj = O(h q+1 ). Since there is no particular

13 Runge{Kutta methods on Lie groups 13 metric on G, we must modify this denition. If we let h vary, ^y 1 will trace out a curve ^y h 2 G. To this curve there corresponds a curve ^ h 2 g via Lemma 2.1. Our denition of order is based on matching the Taylor series of this curve with the Taylor series of the curve t 2 g derived from the analytical solution: Definition 3.1. Let ^y h = ^y 1 be the numerical solution after one step of Algorithm 3.1, considered as a function of h. Let ^y t be the analytical solution of (2.40) and let ^ h and t be the corresponding curves in g given by Lemma 2.1: then the method has order q if ^ h = L 0^y?1 h t = L 0 y?1 t (^y 0 h) (y 0 t ) (3.1) (i) = ^ (i) for i = 0; 1; : : :; q? 1. A very convenient way of analyzing the order of Algorithm 3.1 is to consider it as a 1-stage method on a s-fold product manifold: Let G s = G G, f s = f f, let g s be the Lie algebra of G s and G s its enveloping algebra. We make the identication g s R s g as in Section (2.6) and let A s = AI, b s = b T I. The rst step of Algorithm 3.1 can then be written as: (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) p s = (p; p; : : :; p) 2 G s u = ha s k ~u = (u; k) k = f s (p s exp(~u)) =? R exp ~u f s (ps ) v = hb s k ~v = (v; k) ^y 1 = pexp(~v) = R exp ~v (p) Where u; ~u; k 2 g s, v; ~v 2 g and ^y 1 2 G. In the following analysis, we will let derivations such as ~v 0 and ~v (i) be with respect to h. Lemma 3.1. Let ^ h 2 g be dened in Denition 3.1, let ~z h = exp(~u) 2 G s and let ^ h = L 0 ~z?1 (~z 0 ) 2 h g s. Then h ^ h = dexp ~v (~v 0 ) ^ h = dexp ~u (~u 0 ) Proof: ^ h = L 0 exp(~v)?1l 0 p?1(pexp(~v)) 0 = L 0 exp(?~v)exp 0 (~v; ~v 0 ) = dexp ~v (~v 0 ) : The equation for ^ h follows by a similar computation. }

14 14 Hans Munthe-Kaas We will henceforth make the following assumption about the correction functions: Assumption 1. The functions (; ) in (3.4) and (3.7) are chosen such that ^ (i) = u (i+1) for i = 0; 1; : : :; q? 2, ^ (i) = v (i+1) for i = 0; 1; : : :; q? 1. Note 3.1. If G is Abelian, i.e. if ad(u) = 0 for all u 2 g, then we see from (2.9) and Lemma 3.1 that ^ (i) = ~u (i+1) and ^ (i) = ~v (i+1). Thus in this case the assumption holds if we choose ~u = u and ~v = v. Theorem 3.2. Under Assumption 1, the Lie{Butcher series for the numerical solution ^y h is given by the following recursion for ^ (i)? : (3.9) k (i) = B i A s k (0) ; 2A s k (1) (i?1) ; : : : ; ia s k (3.10) ^ (i) = (i + 1)b s k (i) ; for i = 0; 1; : : :; q? 1. Proof: From k =? R exp ~u f s (ps ), Lemma 3.1 and eqns. (2.31),(2.38) we get k (i) = B i ^ (0) ; : : : ; ^ (i?1) [f s ](p s ) = B i?^ (0) ; : : :; ^ (i?1) ; and from the Leibniz rule i we get: Thus from Assumption 1, h i hf(h)j h=0 = i i?1 h i?1 f(h)j h=0 and eqns. (3.3),(3.6), v (i) = ib s k (i?1) u (i) = ia s k (i?1) ^ (i) = (i + 1)A s k (i) ^ (i) = (i + 1)b s k (i) which yield (3.9),(3.10). } We will use this recursion to obtain the terms ^ (i), and hence also the order conditions on A i;j and b j. Note the following relation between the bar-operator on G s and on G: qv = (q 1 v; q 2 v; : : : ; q s v) [f; : : : ; f] (p; : : : ; p) = (q 1 v; : : : ; q s v) = qv : Let c = A1, where 1 = (1; 1; : : :; 1) T. We get: k (0) = B 0 () = I s = 1I k (1) = B 0? A s k (0) = (AI)? 1I = A1I= ci k (2) = B 2? A s k (0) ; 2A s k (1) =? A1I? A1I + 2AcI= ccii+ 2AcI k (3) = ccciii+ c(2ac)2ii+ (2Ac)cII+ 3A(cc)II+ 6A 2 ci

15 Runge{Kutta methods on Lie groups 15 ^ (0) = b s k (0) = b T 1I ^ (1) = 2b s k (1) = 2b T ci ^ (2) = 3b s k (2) = 3b T c 2 II+ 6b T AcI ^ (3) = 4b T c 3 III+ 8b T (cac) 2II+ II + 12b T Ac 2 II+ 24b T A 2 ci Comparing this with (2.45)-(2.48), we obtain the order conditions b T 1 = 1 2b T c = 1 3b T c 2 = 1 ; 6b T Ac = 1 4b T c 3 = 1 ; 8b T (cac) = 1 ; 12b T Ac 2 = 1 ; 24b T A 2 c = 1. These are the classical order conditions of Runge{Kutta methods. The crucial observation is that since the products on G s are commutative in the left hand part, the order conditions are the same for elementary dierentials which dier only by the ordering of the branches in the Butcher trees. Hence we must have the same order conditions in the commutative as in the non-commutative case. Theorem 3.3. Let A i;j and b j be coecients which dene a classical Runge{ Kutta method of order q. If Assumption 1 holds, then the Algorithm 3.1 has order q on any Lie group G. 3.1 Computing correction functions We want to study the following problem: Given v h 2 g such that v 0 = 0, let ~v = (v h ) and let ^ h = dexp ~v (~v 0 ). Determine () such that (3.11) ^ (i) = v (i+1) for i = 0; 1; : : :; q? 1. We assume for the time being that v (i) can be computed. We will here show how to compute corrections up to order q = 4. A similar procedure can be employed to arbitrary order, but the computations become more complicated. From (2.9) we get ^ h = dexp ~v (~v 0 ) = ~v 0? 1 2 [~v; ~v0 ] [~v; [~v; ~v0 ]]? 1 24 [~v; [~v; [~v; ~v0 ]]] + : : : We assume that ~v = 0 for h = 0. Using [u; v] 0 = [u 0 ; v] + [u; v 0 ], dierentiating and setting h = 0, we obtain: ^ (0) = ~v (1) ^ (1) = ~v (2) ^ (2) = ~v (3)? 1 2 h~v (1) ; ~v (2)i ^ (3) = ~v (4)? h ~v (1) ; ~v (3)i h ~v (1) ; h~v (2)ii (1) ; ~v

16 16 Hans Munthe-Kaas >From this we nd that the corrections hold up to order 4 provided ~v (1) = v (1) ~v (2) = v (2) ~v (3) = v (3) + 1 hv (1) (2)i ; v 2 ~v (4) = v (4) + hv (1) (3)i ; v There are several ways to construct corrections compatible with these conditions. By dierentiation we check that the following holds: Lemma 3.4. The following functions give corrections up to order q = 4: Order 2 : ~v = v h Order 3 : ~v = v h + h 6 Order 4 : ~v = v h + h 4 hv (1) ; v h i i i hv (1) ; v h + h2 hv (2) ; v h 24 Note 3.2. These equations do not require that v (i) are exactly known, e.g. for order 4, it is sucient to know an O(h 3 ) approximant for v (1) and an O(h 2 ) approximant for v (2). Such approximants can be computed from the k i in the RK scheme. We have u (n) = na s k (n?1) and v (n) = nb s k (n?1), thus from Theorem 3.2 u (1) = A s k (0) = ci u (2) = 2A s k (1) = 2AcI v (1) = b s k (0) = b T 1I= I v (2) = 2b s k (1) = 2b T ci= I Note that the elementary dierentials I;I;II; : : : 2 g, and can be computed to sucient accuracy from k by linear algebra: k = (k 1 ; k 2 ; : : : ; k s ) = k (0) + hk (1) + h2 2 k(2) + O(h 3 ) = 1I+ hci+ h2 2 c 2 II+ 2AcI + O(h 3 ) Let d = Ac. If A denes a 4-stage RK method with A 1;j = 0, j = 1; 2; 3; 4, then we get: 0 B k 1 k 2 k 3 k 4 1 C A = 0 B c 2 c 2 2 2d 2 1 c 3 c 2 3 2d 3 1 c 4 c 2 4 2d 4 10 C AB I hi h 2 2 II h 2 2 I 1 C A + 0 B 0 O(h 3 ) O(h 3 ) O(h 3 ) This equation can be solved for Iand Ito the required accuracy. 1 C A :

17 Runge{Kutta methods on Lie groups Explicit RK-MK schemes of order 3 and 4 The algorithms in this section integrate the general initial value problem (2.40) on a general Lie group. In the matrix Lie group case (2.40) can be written as (3.12) y t 0 = yt f(y t ) ; y 0 = p; where f : G! g. Algorithm 3.2. Explicit 3. order RK-MK Choose the coecients P A i;j and b j of a classical s-stage, 3. order explicit RK s scheme. Let c i = j=1 A i;j. ^y 0 = p for n = 0; 1; 2; : : : I 1 = k 1 = f(^y n ) for i = 2; : : : ; s u i = h P i?1 j=1 A i;jk j k i = f(^y n exp(u i )) end v = h P s j=1 b jk j ~v = v + h 6 [I 1; v] ^y n+1 = ^y n exp(~v) end Algorithm 3.3. Explicit 4. order RK-MK Choose the coecients A i;j and b j of a classical s-stage, 4. order explicit RK scheme. Let c i = P s j=1 A i;j, and d i = P s j=1 A i;jc j. Compute the coecients (m 1 ; m 2 ; m 3 ) by solving the linear system: (m 1 m 2 m 3 ) 0 c 2 c 2 2 2d 2 c 3 c 2 3 2d 3 c 4 c 2 4 2d 4 Then the 4. order RK-MK scheme is given as 1 A = (1 0 0) : ^y 0 = p for n = 0; 1; 2; : : : I 1 = k 1 = f(^y n ) for i = 2; : : : ; s u i = h P i?1 j=1 A i;jk j ~u i = u i + cih 6 [I 1; u i ] k i = f(^y n exp( ~u i )) end I 2 = (m 1 (k 2? I 1 ) + m 2 (k 3? I 1 ) + m 3 (k 4? I 1 ))=h v = h P s j=1 b jk j ~v = v + h 4 [I 1; v] + h2 24 [I 2; v] ^y n+1 = ^y n exp(~v) end

18 18 Hans Munthe-Kaas Right hand forms of these algorithms are obtained by rewriting (3.12) as y t 0 = ~ f(yt )y t ; where ~ f(a) = Ad(a)(f(a)), changing the order of multiplication k i = ~ f(exp( ~u i )^y n ) ; ^y n+1 = exp(~v)^y n and changing the sign of some of the correction terms, according to the sign dierences in eqns. (2.9) and (2.10): Order 2 : ~u = u h i Order 3 : ~u = u h? h hu (1) ; u h 6 i Order 4 : ~u = u h? h hu (1) h i ; u h? 2 hu (2) ; u h : A numerical example To illustrate the algorithms, we choose an example given by Zanna [21]. In this example G = O(n), the space of orthogonal nn matrices and g is the space of skew-symmetric nn matrices. We want to solve (3.12) where f(y) is given (in Matlab notation) as: f(y) = diag(diag(y; +1); +1)? diag(diag(y; +1);?1) ; and the initial condition is the following random orthogonal 55 matrix: rand(`seed'; 0); [y 0 ; r] = qr(rand(5)) : We base our algorithms on the original 4 stage 4. order RK scheme given by the coecients A 2;1 = 1=2; A 3;2 = 1=2; A 4;3 = 1; all other A i;j = 0, B 1 = 1=6; B 2 = 1=3; B 3 = 1=3; B 4 = 1=6 : >From this we compute c 1 = 0; c 2 = 1=2; c 3 = 1=2; c 4 = 1; d 1 = 0; d 2 = 0; d 3 = 1=4; d 4 = 1=2; m 1 = 2; m 2 = 2; m 3 =?1 : The numerical experiments were performed by integrating from t = 0 to t = 1 using constants stepsizes, h = ; ; 1 64 ; : : : ; 1 2. Figure 3.1 shows the global error at t = 1 (measured in 2-norm) versus the stepsize h. RK{MK2 is the RK{MK scheme without any correction functions, RK{MK3 and RK{MK4 are corrected to order 3 and 4. All these methods retain orthogonality perfectly. The line `Classic RK4' is obtained by embedding (3.12) in R n2 and use the classical RK scheme here. In this case orthogonality is gradually lost during the integration.

19 Runge{Kutta methods on Lie groups RK-MK2 global error RK-MK Classic RK RK-MK stepsize Figure 3.1: Global error versus stepsize for RK{MK schemes based on the classical 4 stage 4. order RK scheme. 4 Comments added in proof In the period between the submission and the nal proof reading of this article, there has been an major progress in the area of numerical integration techniques for dierential equations on manifolds. In [16], the methods of this paper are generalized to homogeneous manifolds. In [15], we show how classical RK methods of any order can be modied to a Lie group invariant method of the same order, thus we solve the problem of nding correction functions of arbitrarily high order. There has also been a signicant development of ecient methods specialized at equations of Lie type [10]. It is interesting to note that the new Lie group based methods not only give qualitatively better results than classical integrators (due to exact preservation of the group structure), but in several important examples it is also seen that, since they often have a much smaller error constant, they can beat classical integrators also in terms of accuracy versus oating point operations. We believe that the next few years will bring these new integrators more into the main stream of numerical analysis and software. Acknowledgments. I would like to thank Antonella Zanna for many valuable comments and help with the numerical experiments.

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