Asia Pacific Newsletter 1

ButcherButcher Series Series AA Story Story of RootedRooted Trees Trees and and Numerical Numerical Methods Methods for forEvolution Evolution Equations Equations Robert I McLachlan, Klas Modin, Hans Munthe-Kaas and Olivier Verdier Robert I McLachlan, Klas Modin, Hans Munthe-Kaas and Oliver Verdier

Abstract. Butcher series appear when Runge–Kutta fields on vector spaces. Indeed, let f be a smooth methods for ordinary differential equations are ex- vector field on a vector space V, defining the panded in power series of the step size parameter. Each ordinary differential equation (ODE) term in a Butcher series consists of a weighted elemen- tary differential, and the set of all such differentials x˙ = f (x), (1.1) is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher where x˙ = dx denotes the derivative with respect discovered that rooted trees can also be used to obtain dt the order conditions of Runge–Kutta methods, and he to time t. One way to study (1.1) is to develop found a natural group structure, today known as the the Taylor series of its solutions. Let x(h) be the Butcher group. It is now known that many numerical solution to (1.1) at time t = h subject to the initial methods can also be expanded in Butcher series; these condition x(0) = x . The Taylor series of x(h) in h is are called B-series methods. A long-standing problem 0 has been to characterize, in terms of qualitative fea- 1 x(h) = x(0) + hx˙(0) + h2x¨(0) + . (1.2) tures, all B-series methods. Here we tell the story of 2 ··· Butcher series, stretching from the early work of Cayley, We already know that x(0) = x and x˙(0) = f (x ). to modern developments and connections to abstract 0 0 algebra, and finally to the resolution of the character- The additional terms can be found by repeat- isation problem. This resolution introduces geometric edly applying the chain and product rules. For tools and perspectives to an area traditionally explored example, using analysis and combinatorics. d d x¨ = x˙ = f (x) = f (x)x˙ = f (x) f (x), dt dt 1. From Cayley to Butcher or, relative to a basis in which x = x1e + + xne , 1 ··· n n ∂f i Butcher series are mathematical objects that were x¨i = (x) f j(x), introduced by the New Zealand mathematician ∂xj j=1 John Butcher in the 1960s. He introduced them 1 n where f (x) = f (x)e1 + + f (x)e . Continuing in as part of his study of Runge–Kutta methods, a ··· n this way gives popular class of numerical methods for evolu- tion equations such as initial-value problems for x˙ = f (x), ordinary differential equations, and they remain x¨ = f (x) f (x), ... indispensable in the numerical analysis of differ- x = f (x) f (x) f (x) + f (x)( f (x), f (x)), ential equations. In this article we provide a brief .... x = f (x) f (x) f (x) f (x) + f (x) f (x)( f (x), f (x)) introduction to Butcher series, survey their early + 3f (x)( f (x) f (x), f (x)) + f (x)( f (x), f (x), f (x)), history up to their introduction by John Butcher, . and relate the story of the many connections that . (1.3) have recently been discovered between Butcher Here the kth derivative f (k)(x) of the vector field series and other parts of mathematics, notably f is regarded as a multilinear map Vk V. For algebra and geometry.1 We begin, however, with → example, f ( f , f ) is the vector field on V whose the traditional definition. ith coordinate is Butcher series are intimately associated with n ∂2f i the set of smooth (infinitely differentiable) vector (x) f j(x) f k(x). ∂xj∂xk j,k=1 1This article is not a comprehensive review and is focussed on our own interests. Useful companions to this article are A vector field of the form appearing in (1.3), com- the detailed mathematical review of Butcher series by Sanz- bining f and its derivatives, is called an elementary Serna and Murua [35] and the textbook treatments of Hairer et al. [21, 23]. differential. Using (1.3), the Taylor series (1.2) for

December 2017, Volume 7 No 1 1 Asia Pacific Mathematics Newsletter 2

the solution of (1.1) can be written as in Hairer, Lubich and Wanner [21], and a detailed history in Butcher and Wanner [9]. 1 2 1 3 1 3 x(h) = x + hf + h f f + h f f f + h f ( f , f ) + 0 2 6 6 ··· The first breakthrough paper dates from 1963 (1.4) [5]. Here Butcher found for the first time the where each elementary differential is evaluated coefficients ci of the B-series (1.5) of xn+1 of the at x . Notice that the power of h in each term is 0 Taylor expansion in h of an arbitrary Runge– determined by the multiplicity of f in the elemen- Kutta method. This gave the order conditions tary differential. However, the coefficients 1, 1, for Runge–Kutta methods in complete generality. 1/2, 1/6, 1/6, and so on are not determined by their As previous studies had laboriously expanded corresponding elementary differentials. A Butcher the solutions of particular (e.g. explicit) meth- series, shortly denoted B-series, is a generalisation ods by hand, this was an enormously important of (1.4) allowing arbitrary coefficients, i.e. a formal development. series of the form Butcher did have, however, some precursors. 2 3 B(c, f ):= c0x0 + c1hf + c2h f ( f ) + c3h f ( f ( f )) The most notable example is the paper of Mer- son [32] from 1957. Robert Henry ‘Robin’ Merson 3 + c h f ( f , f ) + (1.5) 4 ··· (1921–1992) was a scientist at the Royal Aircraft where c R. Although presented here in coor- Establishment, Farnborough, UK, who was in- i ∈ dinates, we shall see that Butcher series do not vited along with more senior numerical analysts depend on the choice of basis. to a conference on Data Processing and Automatic Computing Machines at Australia’s Weapons Re- search Establishment in Salisbury, South Aus- 2. Early History tralia.2 It seems like a long way to go for a Butcher series are named in honour of the New conference in 1957. However, the UK was still per- Zealand mathematician John Butcher. In a pub- forming above-ground atomic bomb tests in South lication career spanning (so far) 60 years he has Australia at that time and the Australian govern- written 167 papers and books, all but 18 of them ment was very keen to be a part of the emerging concerned with Runge–Kutta methods and their era. Merson’s work is bound up with one of generalisations. Most of them involve in some the most significant events of 1957, the launch way the fundamental structure that bears his of Sputnik 1 on 4 October 1957, and the tale of name. Butcher series were introduced in a remark- Farnborough’s involvement is told in detail by able series of ten sole-authored papers in the years one of the key participants, Desmond King-Hele, 1963–1972. in his book A Tapestry of Orbits [28]. The short version is that with the aid of a large radio an- A Runge–Kutta method is a numerical ap- tenna hastily erected in a nearby field, and some proximation xn xn+1 of the exact flow of (1.1) → calculations of Robin Merson, within two weeks defined by the following equations in xn, xn+1, they had an accurate orbit for Sputnik 1. This X1, ..., Xν V: ∈ allowed them to estimate the density of the upper ν atmosphere and (after Sputnik 2) the shape of the Xi = xn + h aij f (Xj), = earth. Robin Merson became an expert in prac- j 1 (2.1) ν tical numerical analysis and orbit determination.

xn+1 = xn + h bj f (Xj). Merson’s paper explains clearly the struc- j=1 ture of the elementary differentials f ( f ), f ( f , f ), Here ν is the number of stages of the method etcetera, and, crucially, shows how they are in one-to-one correspondence with rooted trees. He and aij, bj are real numbers parameterising the Runge–Kutta method. Associated with the ab- stract Runge–Kutta method (2.1) are its order con- 2Flight-related research at Farnborough began with the Army Balloon Factory in 1904, which became the Royal Aircraft ditions, equations in aij and bj — one Factory in 1912, the Royal Aircraft Establishment in 1918, equation per elementary differential — that deter- and then the Royal Aerospace Establishment in 1988. It was merged into the Defence Research Agency in 1991 and then mine the order of convergence of the method and into the Defence Evaluation and Research Agency in 1995. its local error. Their derivation has been simplified This was split up in 2001, with Farnborough becoming part of the private company Qinetiq. Desmond King-Hele’s version over the years; a modern exposition can be found of these later developments is recorded at [29].

2 December 2017, Volume 7 No 1 4 McLachlan, Modin, Munthe-Kaas, and Verdier 4 McLachlan, Modin, Munthe-Kaas, and Verdier The first breakthrough paper dates from 1963 [5]. Here Butcher found for the The first breakthrough paper dates from 1963 [5]. Here Butcher found for the first time the coefficients ci of the B-series (1.5) of xn+1 of the Taylor expansion in first time the coefficients ci of the B-series (1.5) of xn+1 of the Taylor expansion in 4 McLachlan, Modin, Munthe-Kaas,h of an and arbitrary Verdier Runge–Kutta method. This gave the order conditions for Runge– Kuttah methodsof an arbitrary in complete Runge–Kutta generality. method. As previous This gave studies the order had conditionslaboriously for expanded Runge– Kutta methods in complete generality. As previous studies had laboriously expanded 4 McLachlan, Modin, Munthe-Kaas,the and solutions Verdier of particular (e.g. explicit) methods by hand, this was an enormously The first breakthrough paper dates from 1963 [5]. Here Butcher foundthe for solutions the of particular (e.g. explicit) methods by hand, this was an enormously important development. first time the coefficients ci of the B-series (1.5) of xn+1 of the Taylor expansionimportant in development. The first breakthrough paper dates from 1963 [5]. Here Butcher found for the h of an arbitrary Runge–Kutta method. This gave the order conditions forButcher Runge–Butcher did didhave, have, however, however, some some precursors. precursors. The The most most notable notable example example is is the first time the coefficients c of the B-series (1.5) of x of the Taylor expansion in Kutta methods in completei generality. As previousn studies+1 had laboriouslypaperpaper expanded of Merson of Merson [32] [32] from from 1957. 1957. Robert Robert Henry Henry ‘Robin’ ‘Robin’ Merson Merson (1921–1992) (1921–1992) was h of an arbitrary Runge–Kutta method. This gave the order conditions for Runge– the solutions of particular (e.g. explicit) methods by hand, this was ana scientist enormouslya scientist at the at Royalthe Royal Aircraft Aircraft Establishment, Establishment, Farnborough, Farnborough, UK, UK, who who was was invited invited Kutta methods in complete generality. As previous studies had laboriously expanded alongalong with with more more senior senior numerical numerical analysts analysts to to a conferencea conference on on Data Data Processing Processing and importantthe solutions development. of particular (e.g. explicit) methods by hand, this was an enormously AutomaticAutomatic Computing Computing Machines Machines at Australia’sat Australia’s Weapons Weapons Research Research Establishment Establishment in importantButcher development. did have, however, some precursors. The most notable example is the Salisbury,Salisbury, South South Australia. Australia.2 It2 seemsIt seems like like a long a long way way to to go go for for a a conference conference in in 1957. 1957. paperButcher of Merson did have, [32] however, from 1957. some Robert precursors. Henry The ‘Robin’ most notableMerson example (1921–1992) is the was However,However, the UKthe UK was was still still performing performing above-ground above-ground atomic atomic bomb bomb tests tests in in South South apaper scientist of Merson at the [32]Royal from Aircraft 1957. Establishment, Robert Henry ‘Robin’ Farnborough, Merson UK, (1921–1992) who was was invited AustraliaAustralia at that at that time time and and the the Australian Australian government government was was very very keen keen to to be be a a part part of alonga scientist with at more the Royal senior Aircraft numerical Establishment, analysts to Farnborough, a conference UK, on Data who Processingwas invited and the emergingthe emerging era. era. Merson’s Merson’s work work is bound is bound up up with with one one of of the the most most significant significant events events Automaticalong with Computingmore senior Machinesnumerical analystsat Australia’s to a conference Weapons onResearch Data Processing Establishment and in of 1957, the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’s Salisbury,Automatic South Computing Australia. Machines2 It seems at Australia’s like a long Weapons way to Research go for a Establishment conferenceof 1957, in in 1957.the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’s involvement is told in detail by one of the key participants, Desmond King-Hele, However,Salisbury, Souththe UK Australia. was still2 It performing seems like a above-ground long way to go atomic for a conference bombinvolvement tests in 1957. in South is told in detail by one of the key participants, Desmond King-Hele, in his book A Tapestry of Orbits [28]. The short version is that with the aid of a However, the UK was still performing above-ground atomic bomb testsin in his South book A Tapestry of Orbits [28]. The short version is that with the aid of a Australia at that time and the Australian government was very keen to belarge a part radio of antenna hastily erected in a nearby field, and some calculations of Robin Australia at that time and the Australian government was very keen to belarge a part radio of antenna hastily erected in a nearby field, and some calculations of Robin the emerging era. Merson’s work is bound up with one of the most significantMerson, events within two weeks they had an accurate orbit for Sputnik 1. This allowed the emerging era. Merson’s work is bound up with one of the most significantMerson, events within two weeks they had an accurate orbit for Sputnik 1. This allowed of 1957, the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’sthem to estimate the density of the upper atmosphere and (after Sputnik 2) the shape of 1957, the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’sthem to estimate the density of the upper atmosphere and (after Sputnik 2) the shape involvement is told in detail by one of the key participants, Desmond King-Hele,of the earth. Robin Merson became an expert in practical numerical analysis and orbit involvement is told in detail by one of the key participants, Desmond King-Hele,of the earth. Robin Merson became an expert in practical numerical analysis and orbit in his book A Tapestry of Orbits [28]. The short version is that with thedetermination. aid of a in his book A Tapestry of Orbits [28]. The short version is that with thedetermination. aid of a large radio antenna hastily erected in a nearby field, and some calculations ofMerson’s Robin paper explains clearly the structure of the elementary differentials f ( f ), large radio antenna hastily erected in a nearby field, and some calculations of Robin ( ) Merson, within two weeks they had an accurate orbit for Sputnik 1. ThisMerson’sf allowed( f , f ), paper etcetera, explains and, crucially, clearly the shows structure how they of the are elementary in one-to-one differentials correspondencef f , Merson, within two weeks they had an accurate orbit for Sputnik 1. This allowed them to estimate the density of the upper atmosphere and (after Sputnikf ( 2)fwith, thef ), shape rooted etcetera, trees. and, He crucially, also introduces shows various how they basic are operations in one-to-one on rooted correspondence trees. This de- them to estimate the density of the upper atmosphere and (after Sputnik 2) the shape of the earth. Robin Merson became an expert in practical numerical analysiswithvelopment, rooted and orbit trees. perhaps He also regarded introduces initially various as a basic bookkeeping operations device on rooted for finding trees. and This keep- de- of the earth. Robin Merson became an expert in practical numerical analysis and orbit velopment,ing track perhaps of the regardeddifferent terms, initially has as over a bookkeeping time become device central for to finding the combinatorial and keep- determination.determination. ing trackAsiaand Pacific algebraic of the Mathematics different study of Newsletter terms,B-series. has over time become central to the combinatorial Merson’s paper paper explains explains clearly clearly the the structure structure of ofthe the elementary elementary differentials differentialsf ( f )f, ( f ), 3 and algebraic The rooted study trees of B-series.T and their associated elementary differentials F (T ) are ff ((ff, f ), etcetera, etcetera, and, and, crucially, crucially, shows shows how how they they are are in one-to-onein one-to-one correspondence correspondence The rooted trees T and their associated elementary differentials F (T ) are also introduceswithwith rooted various trees. trees. He basicHe also also operations introduces introduces various on various rooted basic basic operations operationsbecome on central rootedon rooted trees. to trees.the This combinatorial This de- de- and algebraic trees. Thisvelopment,velopment, development, perhaps perhaps regarded regarded perhaps initially initially regarded as as a bookkeepinga ini- bookkeepingstudy device device of for B-series. for finding finding and and keep-T keep-= /0 , , , , , , , , , ... , inging track track of of the the different different terms, terms, has has over over time time become become central central to the to the combinatorial combinatorial tially as a bookkeeping device for finding and The rooted trees T =and/0 , their , associated, ele-, , , , , , ... , andand algebraic study study of of B-series. B-series. keeping track of the different terms, has over time mentary differentials F (T )=are x, f , f ( f ), f ( f ( f )), f ( f , f ), f ( f , f , f ), f ( f , f ( f )), f ( f ( f , f )), f ( f ( f ( f ))), ... . The rooted trees treesTT andand their their associated associated elementary elementary differentials differentialsF (TF)(Tare) are F (T )= x, f , f ( f ), f ( f ( f )), f ( f , f ), f ( f , f , f ), f ( f , f ( f )), f ( f ( f , f )), f ( f ( f ( f ))), ... . Merson introduces a method for carrying out the required Taylor series expan-  T == /0/0, , , ,,, , ,, , , , , , , , , , , , , Mersonsions, , in introduces elementary, , ..., a... differentials method, ,,... , for and carrying gives an out example the required of a 4th Taylor order series Runge–Kutta expan- ∅ method he derived. However, the actual expansions, although greatly simplified by sions in elementary differentials   and gives an example of a 4th order Runge–Kutta F (T )== xx,,ff,,f f( (ff),),f f( (f f((ff)),)),f f ((ff,,f ),f ),f f ( (f ,f ,f ,ff,),f )f,f( f(,ff,(f f()),f ))f,(f f(f( f(,ffthe,)),f )) usef,(f f of((f elementaryf(f( (f ))),f )))..., ... differentials. . and rooted trees, are still carried out term by term. F (T )= x, f , f ( f ), f ( f ( f )), f ( f , f ), f ( f , f , f ) , f ( f , f ( f )) ,methodf ( f ( f he, f )) derived., f ( f ( However,f ( f ))), ... the. actual expansions, although greatly simplified by the use2 of elementary differentials  and rooted trees, are still carried out term by term. Merson introduces a method for carrying out Interestingly enough, MersonFlight-related does research cite at Cayley. Farnborough began with the Army Balloon Factory in 1904, which be- Merson introduces introduces a a method method for for carrying carrying out out the the required required Taylor Taylor series series expan-came expan- the Royal Aircraft Factory in 1912, the Royal Aircraft Establishment in 1918, and then the Royal the requiredsions in Taylor elementary series differentials expansions and in gives elemen- an exampleHowever, of a 4th orderfrom Runge–Kutta the2 context,Flight-relatedAerospace it Establishment is research not clear at in Farnborough 1988. that It he was began merged with into the the Army Defence Balloon Research Factory Agency in 1904,in 1991 which and then be- sions in elementary differentials and gives an example of a 4th order Runge–Kutta tary differentialsmethod he andderived. gives However, an example the actual of expansions, a 4th actually although laid greatly eyes simplifiedcame on Cayley’sinto the Royalby the Defence Aircraft paper. Evaluation Factory He writes, and in 1912, Research the Agency Royal Aircraft in 1995. Establishment This was split upin 1918,in 2001, and with then Farnborough the Royal method he derived. However, the actual expansions, although greatly simplifiedbecoming by part of the private company Qinetiq. Desmond King-Hele’s version of these later developments the use of elementary differentials and rooted trees, are still carried out termAerospace by term. Establishment in 1988. It was merged into the Defence Research Agency in 1991 and then order Runge–Kuttathe use of elementary method differentials he derived. and However, rooted trees, are stillA formula carried for outinto the term the numberis recordedby Defence term. of at Evaluation trees [29]. ofa and given Research Agency in 1995. This was split up in 2001, with Farnborough the actual2 expansions, although greatly simplified becoming part of the private company Qinetiq. Desmond King-Hele’s version of these later developments Flight-related research at Farnborough began with the Army Balloonorder Factory was in discovered 1904,is recorded which by be- at CAYLEY [29]. [our [12]] 2 Flight-related research at Farnborough began with the Army Balloon Factory in 1904, which be- by the usecame of the elementary Royal Aircraft Factory differentials in 1912, the and Royal rooted Aircraft Establishmentand in quoted 1918, and by then ROUSE–BALL. the Royal . . trees, arecameAerospace still the carried Royal Establishment Aircraft out term Factoryin 1988. by in It term. 1912,was merged Hethe Royal did into not Aircraftthe Defence Establishment Research Agency in 1918, in and1991 then and the then Royal Aerospaceinto the Defence Establishment Evaluation in and 1988. Research It was Agency merged in into 1995. the This Defence wasThis split Research was up in 2001, probably Agency with inFarnborough 1991 the and original then 1892 edition of have theintobecoming coefficients the Defence part ofthe Evaluation of private all elementary company and Research Qinetiq. differentials Agency Desmond in 1995. King-Hele’s This was version split of up these in 2001, later developments with Farnborough at once,becomingis as recorded Butcher part at [29]. of achieved. the private company Qinetiq. Desmond King-Hele’sRouse version Ball’s offamous these later bookdevelopmentsMathematical Recreations is recorded at [29]. , as later editions included Coxeter as As it happens, the required mathematics and and Essays coauthor. This first edition contains just one page structures had already been discovered a century on trees, stating Cayley’s formulae for the number earlier by Arthur Cayley in 1857 [12] (see Fig. 2.2). of trees. Now this same section of Rouse Ball also This is the actual discovery of the objects called discusses the famous Knight’s Tour problem, an trees (connected, cycle-free graphs). In popular astonishingly long-lived problem dating from an treatments of , the development of Arabic manuscript of 840 AD. For example, there graph theory is closely linked with recreational were three articles on Knight’s Tours published in mathematics (the bridges of Konigsberg)¨ and with the Mathematical Gazette in 1956 alone. This prob- chemistry (Cayley’s enumeration of alkanes and lem became a life-long interest of Merson’s, who other families of molecules). One common inter- published tours in 1974 and 1999 (posthumously, pretation of the story is that Cayley introduced in Games and Puzzles magazine, from letters writ- the trees as a purely abstract structure and 17 ten in 1990–91) that are still in many cases the years later — behold the power of mathematics! — best known tours. Although Merson stated [27] found that he could use them to count molecules. that he first became interested in the problem in However, Cayley actually needed trees for exactly 1972, it is not unlikely that in 1957 he rediscov- the purpose we are using them here — to keep ered trees independently because, like Cayley, he track of how vector fields interact when applied needed them, and from his interest in recreational repeatedly to one another — and this purpose was mathematics remembered Rouse Ball’s discussion then forgotten for a hundred years. As the need of Cayley without ever chasing it up. for better numerical integration methods arose John Butcher, at that time a PhD student in towards the end of the 19th century, the required physics at the University of Sydney, was actually tools for a complete theory were indeed already present at Robin Merson’s talk in 1957, but says there, but they had been forgotten. [4] that he did not understand it at all. However, As Frank Harary wrote [24], the seed was planted there. To return to Butcher’s In very many cases and in disciplines in the 1963 paper, he closes with the following statement: physical sciences, the social sciences, com- It happens that this situation is capable of puter science, and the humanities, graphs extensive generalisation and, for example, frequently occur as a natural, useful, and in- keeping this same value ν = 3 it is possible tuitive mathematical model. The consequence to satisfy the 37 conditions necessary for a is that those investigators who were not aware sixth order process. Similarly for any value of the existence of graph theory as a study in of ν a process of order up to 2ν is possible. It its own right were led to rediscover it in order is intended that details of such processes will to apply it. be discussed in a later publication.

December 2017, Volume 7 No 1 3 Asia Pacific Mathematics Newsletter 4

An algebraic theory of integration methods in 1972 [7] — submitted in 1968 — in which John Butcher introduced what is now called the Butcher group.

The B-series (1.5) with c0 = 1 correspond formally to diffeomorphisms close to the flow of f , and the Butcher group operation arises from a product of Fig. 2.1. Merson’s [32] 1957 diagram of rooted trees repre- rooted trees that corresponds to the composition senting elementary differentials, and (bottom) an example of a product of trees, in this case the pre-Lie product explained in of these diffeomorphisms. Sec. 4. To give an example of the group operation of the Butcher group, consider the B-series

α := x0 + hf (x0).

This is associated with the map x x := x + 0 → 1 0 hf (x0) of the forward Euler method. The compo- sition of this map with itself (i.e. two steps of forward Euler) is the map

x x + hf (x ) Fig. 2.2. Cayley’s [12] 1857 diagram of rooted trees represent- 0 →1 1 ing elementary differentials. = x0 + hf (x0) + hf (x0 + hf (x0))

1 2 This was an announcement of Butcher’s dis- = x + hf + h f + hf f + h f ( f , f ) 0 2! covery of the family of Gauss Runge–Kutta meth- 1 3 ods and the first hint of extra structure contained + h f ( f , f , f ) + 3! ··· within the Runge–Kutta order conditions. Meth-  2 1 3 ods with 3 stages have 12 free parameters (aij = x + 2hf + h f f + h f ( f , f ) 0 2! and bj for i, j = 1, 2, 3) and Butcher was extremely 1 4 + h f ( f , f , f ) + . excited to discover that there were values of the 3! ··· parameters that satisfied not just the 8 conditions The last line is the B-series of the Butcher product for order 4, and the 17 conditions required for αα. order 5, but even the 37 conditions required for 1 The inverse α− of the B-series α is the series order 6! He recalls running through the empty associated with the inverse map x1 x0. This map corridors of the mathematics department at the → is one step of backward Euler with time step h. University of Canterbury, where he was then − Its B-series is lecturing, desperately trying to find someone to 2 3 1 understand and to share the excitement [4]. He x hf + h f f h f f f + f ( f , f ) 0 − − 2 fulfilled his intention to publish the details in his  4 1 very next paper [6]. + h f ( f , f , f ) + f f f f + f ( f , f f ) 6 One approach taken by Butcher to approach 1 the structure of the order conditions, suggested + f ( f ( f , f )) + . 2 ··· by this discovery, was to introduce certain simpli-  fying assumptions. These became the cornerstone The coefficient of any elementary differential in of the construction of the efficient high-order ex- these series can be found using simple combina- plicit integrators that are used today. However, the torial operations on trees. source of these simplifying assumptions remained This paper [7] aroused an interest that lead mysterious; only very recently has their algebraic to a crucial event. In Innsbruck, the 28-year- origin been explained [30]. This has allowed them old dozent Gerhard Wanner was studying John to be embedded in systematic families and further Butcher’s early papers and his hard-to-understand reduced the number of stages needed at high preprint [7]. In 1970 the University of Innsbruck order. We take this as further evidence that after was celebrating its 300th anniversary and asked 50 years Butcher’s vision is alive and well. each professor to invite a guest lecturer. Wanner’s This initial intensely creative and productive professor, Wolfgang Grobner,¨ asked Wanner for period came to a head with the publication of a suggestion, and so John Butcher was invited.

4 December 2017, Volume 7 No 1 Asia Pacific Mathematics Newsletter5

Ernst Hairer, who had been Wanner’s best fresh- In our view they have held up pretty well. man analysis student the year before, attended the The first widely-acknowledged division of ODEs lectures. In Wanner’s words [37], “In my opinion, in numerical analysis was into stiff and nonstiff at that time, nobody in the world made the necessary equations. Implicit Runge–Kutta methods turned efforts to understand Butcher’s papers, except Ernst. out to be ideal for stiff equations and explicit He then explained them to me, and I tried to put them ones for nonstiff. With the advent of symplectic in a more understandable form,” and in Butcher’s integrators for Hamiltonian systems, that preserve words [8], “This led to my own contribution being a quadratic conservation law on first variations of recognised, through their eyes, in a way that might solutions, Runge–Kutta methods were found to be otherwise not have been possible.” In 1974 Hairer and suitable too. New classes of methods have been Wanner [22] introduced both Butcher series and the introduced that have features that Runge–Kutta term Butcher group; they also clearly demonstrate methods do not, such as exponential integrators the uses of the series for much more than Runge– like Kutta methods. In Butcher [7], the group elements ez 1 x + = x + φ(hf (x ))hf (x ), φ(z) = − , (3.1) are functions from rooted trees to the reals, such as n 1 n n n z those functions induced from (traditional and con- which can beat implicit Runge–Kutta methods on tinuous stage) Runge–Kutta methods; in Hairer some stiff equations, and the AVF (Average Vector and Wanner [22] the primary objects are the B- Field) method series (1.5) themselves, which obey the group law 1 found by Butcher. x + = x + f (ξx + + (1 ξ)x ) dξ (3.2) n 1 n n 1 − n These discoveries triggered a period of huge 0 1 development in numerical methods for evolution that preserves energy H(x) when f = J− H is a ∇ equations. The subsequent modern history of the Hamiltonian vector field. Both (3.1) and (3.2) have area has been reviewed extensively [9, 21, 23, 35]. expansions in B-series. Here we confine ourselves to some remarks as to On the other hand, some methods such as the the role and significance of Butcher series. leapfrog or Stormer–Verlet¨ method, widely used in molecular dynamics and in video game engines 3. How Important are Butcher Series? for systems of the form x¨ = V(x), do not have −∇ Many areas of inquiry show a tendency to divide B-series — indeed they are not even defined for all adherents into ‘lumpers’ and ‘splitters’. For ex- first order systems x˙ = f (x) — and should certainly ample, in taxonomy, lumpers prefer to name few not be discarded on that account. Our view is lump species, splitters many. Lumpers emphasise sim- if you can, but split if you must. ilarity, splitters emphasise difference. Numerical In fact some would say that there is no practical analysis, like most parts of mathematics, shows reason for preferring methods with a B-series and a gradual tendency over time towards splitting, that the whole concept is merely a mathematical as the true differences between instances are ap- abstraction or (perhaps) convenience. However, preciated and exploited. Thus structure-preserving note that (1.5) lumps not only ODEs, but also nu- methods have been developed for finer and finer merical methods. A very large class of numerical divisions of matrices, differential equations and so methods for ODEs are represented by (1.5). Even on, that, by restricting the problem class, are able before getting to the question of what the posses- to offer superior performance. Iserles [25] alludes sion of a B-series confers on a numerical method, to this when he compares ordinary differential the lumping of numerical methods by B-series equations to Tolstoy’s happy families, that (‘per- presents a fairly rare opportunity in computational haps’, Iserles cautions) all resemble each other, science. All too often one analyzes the complexity while each partial differential equation is unhappy or behaviour of a particular algorithm, or perhaps in its own way. Indeed, a mighty strength, and also of a small class. Meaningful lower bounds for a potential weakness, of Runge–Kutta methods complexity or behaviour over all algorithms are and of B-series is that they treat all ODEs in a almost never obtained. One should not miss the uniform way. They are an extreme example of opportunity given by B-series to better understand lumping. One might wonder if they are perhaps an infinite-dimensional set of methods, without too extreme. Do they over-lump ODEs? regard to particular details of the method.

December 2017, Volume 7 No 1 5 4 McLachlan, Modin, Munthe-Kaas, and Verdier 4 McLachlan, Modin, Munthe-Kaas, and Verdier 4 4The first breakthrough paper dates fromMcLachlan, 1963McLachlan, Modin, [5]. Here Munthe-Kaas, Modin, Butcher Munthe-Kaas, and found Verdier for and theVerdier first time the coefficients ci of the B-seriesThe first (1.5) breakthrough of xn+1 of paper the Taylor dates expansion from 1963 in [5]. Here Butcher found for the Theh of first anThe breakthrough arbitrary first breakthrough Runge–Kutta paper dates paper method.first from timedates 1963 This the from coefficientsgave [5]. 1963 Here the [5]. order Butcherc Herei of conditions the Butcher found B-series for for found the (1.5) Runge– for of thexn+1 of the Taylor expansion in first timeKuttafirst the methods time coefficients the in coefficients completec of the generality.c B-seriesofh theof an B-series (1.5)As arbitrary previous of x (1.5) Runge–Kutta studiesof of thex hadTaylorof laboriously the method. expansion Taylor This expansion expanded in gave the in order conditions for Runge– Asia Pacific Mathematics Newsletter i i 6 n+1 n+1 h of anthe arbitraryh solutionsof an arbitrary Runge–Kutta of particular Runge–Kutta method. (e.g. explicit)Kutta method. This methods gave methods This the in gave ordercomplete by hand, the conditions order generality. this conditionswas for an AsRunge– enormously previous for Runge– studies had laboriously expanded important development. the solutions of particular (e.g. explicit) methods by hand, this was an enormously Kutta4 methodsKutta4 methods in complete in complete generality.3 generality. As previousMcLachlan, As previous studies Modin, had studiesMcLachlan, laboriously Munthe-Kaas, had Modin,laboriously expanded and Munthe-Kaas, Verdier expanded and Verdier Several times, new numerical methods have thean example solutionstheButcher of solutions of a pre-Lie particular did have,of algebra. particular (e.g. however, explicit)Another (e.g.important some explicit) methods example precursors. development. methods by hand, The by this most hand, was notable this an enormously was example an enormously is the been reflected in the discovery of new structure importantis thepaper linearimportant development.of combination Merson development. [32] of rooted from 1957. trees, whereRobertButcher the Henry did have, ‘Robin’ however, Merson some (1921–1992) precursors. was The most notable example is the 1 The first breakthroughThe first paper breakthrough dates from paper 1963 dates [5]. from Here 1963 Butcher [5]. Herefound Butcher for the found for the within B-series. For example, if f = J− H for some pre-Lie product is given by grafting: for two trees ∇ firstButcher timea scientist theButcher did coefficientsfirst at have, thetime did however,Royal the have,ci coefficientsof Aircraft however,the some B-seriespaper Establishment, precursors.c somei of (1.5)of the Mersonprecursors. B-seriesof Thexn Farnborough,+ most[32]1 of (1.5) The thefrom notable mostof Taylor 1957.xn UK,+ examplenotable1 expansionof Robert who the example was isTaylor Henry the in invited expansion is ‘Robin’ the Merson in (1921–1992) was H and J, where JT = J defines a symplectic struc- τ and τ the pre-Lie product τ τ is computed − hpaper1 of analong of arbitrarypaper2 Merson withh ofof more Runge–KuttaMerson[32] an arbitraryfromsenior [32] 1957. numerical from method.1 Runge–Kutta Robert2 1957.a scientist analystsThis Henry Robert gavemethod. at to ‘Robin’ Henry the the a conference RoyalThis order ‘Robin’Merson gave conditions Aircraft on the Merson(1921–1992) Data order Establishment, for Processing (1921–1992) conditions Runge– was andFarnborough, for was Runge– UK, who was invited τ ture on the vector space V, then f is Hamiltonian byKuttaa scientist attachingAutomatic methodsa scientist at the theKutta in Computingroot Royal complete at methods of the Aircraft1 Royalwith generality. Machines in an complete Aircraft Establishment, edgealong As at to Establishment, Australia’s previousgenerality. each with of Farnborough, more studies AsWeapons senior previous Farnborough, had numerical UK, laboriouslyResearch studies who UK, wasanalystsEstablishment had expanded who laboriouslyinvited was to a invited conference in expanded on Data Processing and τ 2 and energy preserving and we can ask which B- thealong nodes solutionsSalisbury, withalong of more2 ofthe withand South particular senior solutions adding more Australia. numerical senior (e.g.all of these particular explicit) numericalIt analysts terms seemsAutomatic methods(e.g. together like analysts to aexplicit) a conference longComputing by to hand,way a methods conference to thison Machines go Data forwas by a hand, on Processing an conference at Data enormously Australia’s this Processing was and in an 1957. Weapons enormously and Research Establishment in 2 series have these properties. The trivial B-series importantAutomatic(see Fig.However,Automatic 2.1). development. Computing Theimportant the pre-Lie Computing UK Machines was development. algebra still Machines perspective at performing Australia’sSalisbury, at Australia’s of above-ground WeaponsB- South Australia. Weapons Research atomic ResearchIt Establishment seems bomb like tests Establishment a in long in South way in to go for a conference in 1957. 2 2 B( f ) = c1f are the only ones which are both Salisbury,seriesButcherAustralia wasSalisbury, South promoted did at have, Australia. thatButcher South by time however, Calaque,Australia. did andIt seems have, the some Ebrahimi-Fard, AustralianIt however,likeHowever, precursors. seems a long like somegovernment the way a The long UKprecursors. to most gowas way for was notable stillto a goveryconference The performing for example keenmost a conference to notable in be above-ground1957.is athe part example in 1957. of is atomic the bomb tests in South Hamiltonian and energy-preserving. At first sight paperandHowever, Manchonthe ofHowever, emergingMerson thepaper UK [10]. [32]the era.was A of fromUK fundamental Merson’s Mersonstill was 1957. performing [32]still work Robert result, performingfromAustralia is above-ground bound Henry 1957. which at up above-groundRobert ‘Robin’ that with time atomic one Henry Merson and of thebomb atomic‘Robin’ the (1921–1992) most Australian tests bomb significant Merson in Southtests government was (1921–1992) events in South was was very keen to be a part of it is surprising that the first nontrivial B-series, awasAustralia scientist essentiallyof 1957,Australia at at that thea the known scientist time Royal atlaunch that and already Aircraft attime of the the Sputnik Australianand toRoyal Establishment, Cayley the 1the Aircraft Australian on emerging government in 4 October 1857, Establishment, Farnborough, government era. 1957, was Merson’s very and UK,wasFarnborough, thekeen work verywho tale to is ofbe keenwas bound Farnborough’s a UK, invitedpart to up be whoof with a part was one of invited of the most significant events f f , is neither Hamiltonian nor energy-preserving. alongbutthe emergingwhichinvolvement withthe has more emerging era. beenalong senior Merson’s is revisited with told era. numerical morein Merson’s work in detail a senior modern is analysts by bound workof numericalone 1957, algebraic is up of to bound with the a the conference analysts key launchone up with participants, of the ofto one on Sputnika most conference ofData the significant Desmond Processing 1 most on on4 significant October eventsData King-Hele, and Processing 1957, events and and the tale of Farnborough’s At the next order, f f f is energy preserving and Automaticsettingof 1957,in by hisof the Chapoton Computing1957, book launchAutomatic theA ofTapestry andlaunch MachinesSputnik Computing Livernet of of Sputnik1 Orbits aton in Australia’s 4 Machinesinvolvement 2001 October[28]. 1 on [13], 4 The OctoberWeapons 1957,at is Australia’sshort is told and 1957, version Research in the detail Weapons tale and is of the Establishmentthat by Farnborough’stale oneResearch with of of the Farnborough’s the aid Establishment in key of participants, a in Desmond King-Hele, f ( f , f ) 2f f f is Hamiltonian. The spaces of such thatinvolvement thelarge spaceinvolvement radio is of told all antenna trees in is detail told with hastily2 in bythe detail erected one graftingin of by in his the2 one product a book nearbykey of theparticipants,A field, Tapestry key participants, and ofsome Desmond Orbits calculations Desmond[28]. King-Hele, The of King-Hele, short Robin version is that with the aid of a − Salisbury, SouthSalisbury, Australia. SouthIt seems Australia. like aIt long seems way like to go a long for a way conference to go for in a 1957. conference in 1957. B-series have been completely described [15]. isHowever,in histheMerson, bookfreein the his pre-LieAHowever, UKwithin Tapestrybook was algebraA two Tapestry still of the. weeks Orbits This performing UK of meanstheywas[28]. Orbitslarge still had The above-ground thatperforming[28].an radio short thisaccurate The antenna version short above-ground atomicorbit hastily is version for that bomb erectedSputnik with is testsatomic that the in 1. with ain aid This nearby bomb South of the allowed a testsaidfield, of in and a South some calculations of Robin structureAustralialargethem radiolarge ‘knows at to antenna that estimate radioAustralia time all antennahastily there and the at density isthethat erected hastily to Australian timeknow’ of in erected andthe aMerson, about nearby upper the government in Australianbasic a atmospherewithinfield, nearby and two was field, government someweeks andvery and calculations (after keen some they was Sputnik tohad calculations be very ofana part2) Robin keenaccurate the of of shapeto Robin be orbit a part for of Sputnik 1. This allowed algebraicMerson,of theMerson, within properties earth. two withinRobin ofweeks pre-Lie Merson two they weeks algebras, became had theythem an an accurate andhad toexpert estimate anany accurate inorbit practical the for densityorbit Sputnik numerical for of Sputnik 1. the This analysis upper allowed 1. atmosphere This and allowedorbit and (after Sputnik 2) the shape 4. Algebraic Characterisations the emerging era.the Merson’s emerging work era. Merson’s is bound work up with is bound one of up the with most one significant of the most events significant events algebraicofthem 1957,determination. to estimatethem the computation launch toof estimate the1957, of density Sputnik thewhich the launch of densityrelies the 1 on ofupper of4 onlyofSputnik October the the atmosphere on upper earth. 1 the 1957, on atmosphere Robin 4 andOctober and Merson (after the and 1957,tale Sputnik became (after of and Farnborough’s 2)Sputnik an the the expert tale shape 2) of in the Farnborough’s practical shape numerical analysis and orbit The topic of B-series can be approached from pre-Lieinvolvementof the earth. relationshipofMerson’s the Robin isinvolvement earth. told paperMerson can inRobin detail be explains became expressedMerson is by told one clearly inan becamedetermination.as ofdetailexpert a the compu- an structurebyin key expert practical one participants, of in of the practical thenumerical key elementary Desmond participants, numerical analysis differentials King-Hele, analysisand Desmond orbit andf ( f King-Hele, orbit), f ( f , f ) many different points of view; topics in numer- tationindetermination. his on book determination. trees.A,in etcetera,TapestryIt alsohis book means and, ofA Orbits that crucially, Tapestry any[28]. example ofshows TheMerson’s Orbits short how of a[28]. theypaper version The are explains is shortin that one-to-one version clearly with the the iscorrespondence aid thatstructure of with a of the the aid elementary of a differentials f ( f ), with rooted trees. He also introducesf ( f various, f ) basic operations on rooted trees. This de- ical analysis, geometry and abstract algebra are concretelargeMerson’s radio pre-Lie antennaMerson’s paperlarge algebra explains hastily radio paper can antenna erected explains clearly be realised hastily in the clearly a structure nearby as erected a the quo-, field, etcetera,structure of in the a and nearby elementary and,of some the field, crucially, elementarycalculations differentials and someshows differentials of calculations Robin howf ( f ) they, f are( off ) in, Robin one-to-one correspondence f ( f ,velopment,f ) f ( f , f ) perhaps regarded initiallywith rooted as a bookkeeping trees. He also device introduces for finding various and basic keep- operations on rooted trees. This de- connected via B-series. The fundamental algebraic tientMerson, of, the within etcetera, freeMerson, two, pre-Lie etcetera, and, weekswithin crucially, algebra and, they two crucially,had with shows weeks an some how accurate shows they ideal they had how orbit are an they in accurate for one-to-one are Sputnik in orbit one-to-one 1. correspondence for This Sputnik allowed correspondence 1. This allowed with rootedingwith track trees. rooted of He the trees. also different introduces He also terms, introduces variousvelopment, has over basic various time perhapsoperations basic become regardedoperations on central rooted initially on to trees. rootedthe ascombinatorial This a trees. bookkeeping de- This de- device for finding and keep- structure of a pre- unifies three seem- (thatthem is,to estimate as treesthem with the to density some estimate equivalence of the the upper density relation). atmosphere of the upper and atmosphere (after Sputnik and 2)(after the Sputnik shape 2) the shape velopment,andvelopment, algebraic perhaps study perhapsregarded of B-series. regarded initiallying initiallyas a track bookkeeping as of a the bookkeeping different device for terms, device finding has for and over finding keep- time and become keep- central to the combinatorial ingly very different papers all written in 1963: Thisof the is earth. indeed Robinof a usefulthe Merson earth. result Robin became for Merson computations. an expert became in practical an expert numerical in practical analysis numerical and orbit analysis and orbit ing trackingThe of track the rooted different of trees the different terms,T and has their terms, overand associated has algebraic time over become elementary time study become central of B-series. differentials to central the combinatorial toF the(T combinatorial) are John Butcher’s first paper on Runge–Kutta meth- determination.The correspondencedetermination. between abstract trees and and algebraicand algebraic study of study B-series. of B-series. The rooted trees T and their associated elementary differentials F (T ) are ods [5], Ernest Vinberg’s paper on the geometry concreteMerson’s elements paper inMerson’s explains a given paper clearly pre-Lie explains the algebra structure clearly (e.g. of a the the structure elementary of the differentials elementaryf differentials( f ), f ( f ), f (Thef , f ) rooted, etcetera,Thef trees rooted( f and,, Tf ), treesandcrucially, etcetera, theirT and associatedand, shows their crucially, how associated elementary they shows are elementary in how differentials one-to-one they differentials areF correspondence in( one-to-oneT ) areF (T ) correspondenceare of symmetric cones [36] and Murray Gersten- vector fieldT on=R n/0) , is exactly, , the elementary, differ-, , , , , ... , with rooted trees.with He rooted also introduces trees. He also various introduces basic operations various basic on rooted operations trees. on This rooted de- trees. This de- haber’s work on homology and deformations of ential map of Butcher. The elementary differentialT = /0 , , , , , , , , , ... , velopment, perhapsvelopment, regarded perhaps initially regarded as a bookkeeping initially as a device bookkeeping for finding device and for keep- finding and keep-  algebras [19]. The differential geometric picture mapT =(τ),)=/0 takingT , ,=x, treesf/0, , ,f ,( tof ), vectorf ,( f, ( fields,f )), f , respects(, f , f ), f , ( f ,, f , f ), f ,( f , f, ( f )), f ,( f (,f , f )), f ,( f ( f ,(...f ))),,..., .... , ing trackF T of theing different track of terms, the different has over terms, time has become over central time become to the combinatorial central to the combinatorial  starts with the basic notion of parallel transport of the structure of the pre-Lie product, (τ τ ) = and algebraic studyand algebraic of B-series. study of B-series.F (T1 )=2 x, f , f ( f ), f ( f ( f )), f ( f , f ), f ( f , f , f ), f ( f , f ( f )),f ( f ( f , f )), f ( f ( f ( f ))), ... . vectors, which is infinitesimally described in terms (τ1))= (τx2),)=f wheref (xf ) f thef f( f( trianglef()f ))f (ff on((ff,)) thef ) f leftf( f(, isff,)f ,ff )( f , (f ,f ,f )f (ff ))( f ,ff((ff ))( f ,ff ))( f f( f(,ff())f (ff ()))f (...f ( f ))). ... . F TThe rootedFMersonT, trees, introducesThe T, , rootedand, their a trees method, , associated T and for, their, carrying elementary associated , out , differentials the elementary required, F, Taylor differentials( T ) are series, ,F expan- (T ) are, , ,  of a connection or covariant derivation of vector graftingsions of in trees elementary and on differentialsthe right is andtheMerson covari-gives an introduces example a of method a 4th order for carrying Runge–Kutta out the required Taylor series expan- fields. The connection is a bilinear operation of antMerson derivativemethodMerson introduces he of derived. vector introduces a fields. method However, All a formethod the the carryingsions elementary actual for in carrying elementaryout expansions, the required out differentials thealthough required Taylor greatly seriesand Taylor gives simplified expan- series an example expan- by of a 4th order Runge–Kutta vector fields ( f , g) f g (often written as g) differentialsT = /0 are, , obtainedT =, /0 this , , way., For, example,, , , , ,, ,, ,, ... , , ... , → ∇f sionsthe insions elementary use of in elementary elementary differentials differentials differentials andmethod gives and and an rooted hegives example derived. trees, an example of are However, a still 4th of ordercarried a the 4th Runge–Kutta actual out order term expansions, Runge–Kutta by term. although greatly simplified by which describes the rate of change of g as it is since = ( ) ( ) , we must have that if method hemethod derived. he− derived. However, However, the actualthe theexpansions, use actual of elementary expansions, although differentials although greatly simplified greatly and rooted simplified by trees, are by still carried out term by term. parallel-transported along the flow of f . On the ( ) =)=2f ,Flight-related then ( ())= research=) f ((f at(f ) Farnborough))( (f f))( f,.( Similarly, began) ( )) with( , the(, ,) Army) ( Balloon,( ,( ,)) Factory) ( ( in(, 1904,,( )))) which(( be-( ( (, ))))) ...( .( ( ))) ... . FtheT use ofthe elementaryx use, Ff , offT elementaryf differentials, f xf, f ,f differentials−f , andf , ff rooted ff , andff trees, rooted,ff f areff trees, stillf, f, f carried are f f still ff outf carried,,f termf f f out byff term. termff ,,f byf f term.f f f ff ,, f f f f , Rn came the Royal Aircraft Factory in 1912,2 the Royal Aircraft Establishment in 1918, and then the Royal vector space parallel transport is the obvious all the terms of the B-series can be expressedFlight-related in research at Farnborough began with the Army Balloon Factory in 1904, which be- 2 Aerospace2 Establishment in 1988. It wascame merged the Royal into the Aircraft Defence Factory Research in 1912, Agency the Royal in 1991 Aircraft and then Establishment in 1918, and then the Royal rule, and the corresponding connection is given as termsFlight-relatedMersoninto of the the DefenceFlight-relatedintroduces pre-Lie researchMerson Evaluation product, at a research Farnborough method introduces and andat Research Farnborough for hencebegan carrying a Agencymethod with we began the caninout for 1995. Army with the carrying This the Balloonrequired Army was out splitFactory Balloon Taylor the up in in requiredFactory 2001, 1904,series with which in expan- Taylor1904, Farnborough be- which series be- expan- came the Royalcame theAircraft Royal Factory Aircraft in Factory 1912, the in Royal 1912,Aerospace theAircraft Royal Establishment Establishment Aircraft Establishment in 1988. in 1918, It was and in merged1918, then the and into Royal then the the Defence Royal Research Agency in 1991 and then n i regardsionsbecoming in a elementary B-series partsions asof the an indifferentials private infiniteelementary company expansion and differentials Qinetiq.into gives thein Desmond an a Defence pre- example and King-Hele’s Evaluation gives of an a and4th exampleversion Research order of these of Runge–Kutta Agency a later 4thdevelopments in order 1995. Runge–Kutta This was split up in 2001, with Farnborough ∂g j ∂ AerospaceAerospace Establishment Establishment in 1988. It in was 1988. merged It was into merged the Defence into the Research Defence Agency Research in Agency 1991 and in then1991 and then f g = g( f ) = f . Lie product.is recorded at [29]. becoming part of the private company Qinetiq. Desmond King-Hele’s version of these later developments ∂x j ∂xi intomethod the Defence heinto derived. the Evaluationmethod Defence However, Evaluation he and derived. Research the and AgencyactualHowever, Research inexpansions, Agency 1995. the This actual in 1995. was although expansions, split This up was in greatly split 2001, although up withsimplified in 2001, Farnborough greatly with by Farnborough simplified by i ,j=1 becomingtheAre use of there partbecoming elementary of the otherthe part private use of important differentialsof the company elementary private Qinetiq. company examples and differentialsis Desmond rooted Qinetiq. recorded of trees, King-Hele’spre- Desmond at and[29]. are rooted King-Hele’s still version carried trees, of versionthese are out later still termof these developments carried by later term. outdevelopments term by term. The curvature R and the torsion T are the two Lieis recorded algebrasis at recorded [29]. where at [29]. B-series might play a role? basic invariants of a connection. On flat spaces, There2 Flight-related was a great research2 Flight-related surprise at Farnborough in research the late began at 1990s Farnborough with when the Army began Balloon with the Factory Army in Balloon 1904, whichFactory be- in 1904, which be- such as the above defined connection on Rn, both Christiancame the Royal Brouder Aircraftcame the pointed Factory Royal in Aircraft out 1912, [2] the Factory Royalthat in theAircraft 1912, so- the Establishment Royal Aircraft in 1918, Establishment and then in the 1918, Royal and then the Royal Aerospace EstablishmentAerospace in 1988.Establishment It was merged in 1988. into It thewas Defence merged Researchinto the Defence Agency Research in 1991 and Agency then in 1991 and then R = 0 and T = 0. It can be shown that in this calledinto the Defence Evaluationinto the Defence of and Alain Research Evaluation Connes Agency and and Research in 1995. Dirk This Agency was in split 1995. up in This 2001, was with split Farnborough up in 2001, with Farnborough case the connection satisfies the following pre-Lie Kreimerbecoming part [16] of had thebecoming private the same partcompany of algebraic the Qinetiq. private structure company Desmond Qinetiq. King-Hele’sthat Desmond version King-Hele’s of these later version developments of these later developments relation: Johnis recorded Butcher at [29]. hadis recorded been at studying [29]. in detail in his 1972 paper. Connes and Kreimer had been in- f ( g h) ( f g) h = g ( f h) ( g f ) h. − − 3Also called a Vinberg, Koszul–Vinberg, left-symmetric, or An algebra with a product satisfying this relation- Gerstenhaber algebra. The name reflects the fact that the skew ship is called a pre-Lie algebra. So, the set of smooth product [x, y] := x y y x defines a Lie bracket. However it should be noted that− the pre-Lie relation is not the most vector fields on Rn with the standard connection is general form of a product with this property.

6 December 2017, Volume 7 No 1 Asia Pacific Mathematics Newsletter7 terested in renormalisation processes in quantum remark on other recent geometric developments of field theory and discovered a rich algebraic struc- the theory. ture of trees. Indeed Arne Dur¨ [17] had already Concerning the group structure of B-series, observed in 1986 that Butcher had given rooted Bogfjellmo and Schmeding [1] have recently trees the structure of a Hopf algebra. Rereading proved that the space of B-series is an infinite- Butcher [7] in light of these more recent develop- dimensional with respect to a natu- ments, it is striking how close his perspective is ral Frechet´ topology. Among numerical analysts, to the modern Hopf algebraic view. As Brouder B-series have long been treated as Lie groups commented, “Butcher found an explicit expression without a rigorous justification; the result by for all the operations of the Hopf structure of the Bogfjellmo and Schmeding resolves this and un- algebra of rooted trees.” After Brouder’s work the veils interesting possibilities to apply tools from Fields medallist Alain Connes wrote [16] “We re- infinite-dimensional geometry to the backward gard Butcher’s work on the classification of numerical error analysis of ODE methods. integration methods as an impressive example that The question of characterising geometries by concrete problem-oriented work can lead to far-reaching invariance properties goes a long time back to conceptual results.” Pierre Cartier has also written a the 19th century work of Felix Klein, who in very clear exposition of the significance of pre-Lie his Erlangen program of 1872 raised fundamental algebras and the algebraic origin of the Connes– questions about geometries and symmetries. An Kreimer approach [11] . example is the study of affine geometries as a gen- More recently these algebraic structures appear eralisation of Euclidean spaces. In this geometric in other important areas, such as in stochastic context it is interesting to ask if other geometries processes, where the Rough Paths Theory gives a have algebras describing their connections, such precise meaning to integrating functions along as pre-Lie algebras for affine geometries. Recent highly irregular paths. This theory originated from developments have shown that this is indeed the the work of Terry Lyons and was celebrated by the case. For Lie groups and homogeneous spaces Fields medal awarded to in 2014 there are naturally defined connections which give for his work on regularity structures. Relations rise to post-Lie algebras, and from this we obtain B- between rough paths and B-series have been de- series types of expansions valid for flows evolving veloped in the work of Massimo Gubinelli [20]. on manifolds (‘Lie–Butcher’ series) [33]. Yet an- In a completely different direction, expansions other algebra appears in the context of symmetric in rooted trees can be used to dramatically sim- spaces such as, for example, spheres and Rieman- plify and also to sharpen known results in com- nian spaces with constant curvature. This is an plex dynamics [18] (“this amounts to a novel ap- active area of research, where differential geome- proach to formal linearisation by means of a powerful try, algebraic combinatorics, differential equations, and elegant combinatorial machinery”). computations and applications go hand-in-hand. Considering B-series as an expansion in a (flat and torsion free) connection, we may ask what 5. Geometric Characterisations are the characterising geometric properties of a B-series? A partial answer comes from the ques- Many mathematical objects can be defined in tion of which invertible mappings φ: Rn Rn different ways: axiomatically, constructively, or → preserve the connection . Let φ act on vector by characterising their relationship to another, fields in the ‘natural’ way (i.e., as a differential known, object. The original, and still the tradi- equation transforms under change of coordinates) tional, approach to Butcher series [21] is construc- 1 φ f := (φ) f φ− , where φ is the Jacobian tive. It is motivated by the Taylor series of the · ◦ ◦ matrix. Then it can be shown that φ ( f  g) = exact solution. It starts by constructing the rooted · (φ f )  (φ g) for all vector fields f and g if and trees, most easily done recursively using the oper- · · only if φ(x) = Ax + b is an affine map. However, ation of adding a root to a forest (set of rooted it turns out that this condition is not enough to trees). Then the elementary differentials are de- nail precisely the question of What is a B-series?, fined and associated to the rooted trees, and finally but we shall see towards the answer. Before we it is shown that various objects (Runge–Kutta and explore this issue further in the next section, we other integration methods) can be expanded in

December 2017, Volume 7 No 1 7 Asia Pacific Mathematics Newsletter 8

Butcher series. The algebraic approach of the pre- Table 5.1. Enumeration of rooted and aromatic trees with up to vious section is axiomatic. However, if we recall 10 nodes. n 123 4 5 6 7 8 9 10 the origin of Butcher series in numerical analysis, # rooted trees 1 1 2 4 9 20 48 115 286 719 and note that not all numerical integrators have # aromatic trees 1 2 6 16 45 121 338 929 2598 7261 a Butcher series, it is natural to ask why these

particular combinations, f ( f , f ) and so on, keep series begins coming up. What is special about them? What geometric property characterises those numerical c0x + c1hf 2 integrators that have a Butcher series? + h (c f f + c f f ) 2 3 ∇· 3 A crucial clue is provided in the definition of + h (c f ( f , f ) + c f f f + c f ( f ( f )) 4 5 6 ·∇∇· Runge–Kutta methods, (2.1). Apart from evalua- 2 2 + c7 f f f + c8 f ( f ) + c9 f tr( f )) tion of f , these involve only scalar multiplication ∇· ∇· + and addition — the defining operations of the ··· vector space V. This suggests that Runge–Kutta which may be represented as an element in the methods are defined intrinsically on V and do span of the aromatic trees not depend on the choice of basis. Indeed, as , already mentioned previously in the context of , , pre-Lie algebras, slightly more is true: Runge– Kutta methods (and B-series) are affine-equivariant. , , , , , , Indeed, let, as before, smooth invertible mappings ... φ: V V act on the vector space V and on → vector fields on V in the natural way. Then B-series There are clearly many more aromatic than rooted

with c0 = 1, such as the expansions of numerical trees. The aromatic trees of order n are in 1–1 integrators, obey correspondence with functions from 2, ..., n to { } 1, ..., n , ‘forgetting the labels’, that is, modulo φ B(c, f ) = B(c, φ f ) { } · · permutations of 2, ..., n . (Here the element 1 { } n n for all invertible affine maps φ(x) = Ax+b, A R × , identifies the root.) For example, the aromatic tree ∈ det A 0. Could it be the case that any affine- 2 3 equivariant method has a Butcher series? In other words, does affine-equivariance characterise B- 4 1 series methods? In [34], two of us showed that this is not the case. There are many methods that are affine- is associated with the function 2 1, 3 4, 4 4 → → → equivariant but do not have Butcher series. The and with the (generalised) elementary differential simplest example is the first-order method n ∂ i1 i i i4 f f 2 f 3 f = f ( f )(f ( f )). i2 i3i4 i x1 = x0 + hf (x0)(1 + h( f )(x0)). ∂x 1 ·∇∇· ∇· i1,i2 ,i3,i4=1 Under an affine transformation x φ(x) = Ax + b, The numbers of such ‘shapes of partially defined 1 → f transforms to Af φ− , and the Jacobian f functions’ is given in sequence A126285 in the ◦ 1 1 transforms to A( f φ− )A− . The divergence of f , Online Encyclopedia of Integer Sequences and tab- ◦ 1 namely tr f , transforms to (tr f ) φ− , and the new ulated in Table 5.1.The number of rooted trees, first ◦ 1 term f f transforms to A( f f ) φ− — that is, evaluated by Cayley, are shown for comparison. ∇· ∇· ◦ it is affine equivariant. The apparently terrifying numbers of rooted trees It turns out that any affine-equivariant method were tamed by Butcher. What will happen to the can be expanded in terms of more general objects, even more plentiful aromatic trees? the aromatic series. Combinatorically, these are rep- The existence of the aromatic series shows that resented by ‘aromatic trees’, forests consisting of affine-equivariance of a method is not enough to one rooted tree and any number of directed graphs ensure that it can be expanded in a B-series. What with one cycle (self-loops allowed). The name is else is needed? The second big clue is that Runge– suggested by aromatic compounds, such as ben- Kutta methods are defined without reference to zene, that contain cycles of atoms. An aromatic the dimension of the underlying vector space. It

8 December 2017, Volume 7 No 1 Asia Pacific Mathematics Newsletter9 does not seem to play any role at all. Clearly, at of the factors, leading to the 6 aromatic trees of a minimum, the expansion of the method in each order 3. dimension must have the same coefficients. But The proof of the theorem on affine related- what rules out the aromatic terms like f f ? ness, characterising B-series [31], begins with an ∇· The answer is that these terms do not respect arbitrary affine-related method. Since, in partic- affine-relatedness. Consider two vector spaces V and ular, it is affine-equivariant, it has an aromatic W of possibly different dimension, together with series. Each aromatic tree containing loops is to be an affine map φ: V W, x Ax + b. The knocked out. For each such tree, a special pair of → → vector fields f on V and g on W are said to be affine-related vector fields is constructed such that φ-related if g(Ax + b) = Af (x) for all x V. B-series affine-relatedness of the method means that the ∈ preserve affine-relatedness in the sense that for coefficient of this tree must be zero. For example, any affine φ, if and are φ-related then ( , ) for the tree , associated with f f , the vector f g B c f ∇· is φ-related to B(c, g). In [31] we prove that this fields are f (1) : x˙1 = 1, x˙2 = x2 and f (2) : x˙1 = 1. property characterises B-series: a numerical method These vector fields are related by the affine map (x1, x2) x1. Since f (1) f (1) = 1 and f (2) f (2) = 0, has a Butcher series if and only if it preserves affine- → ∇· ∇· relatedness. this term cannot appear in the expansion of a Preserving affine-relatedness has a fairly direct method that preserves affine-relatedness. physical interpretation. It means that the method To summarise, Butcher series are objects in- is immune to changes of scale, such as changes of trinsically associated to the set of vector fields on units. It means that the method preserves invari- affine spaces of all dimensions, and will show up ant affine subspaces automatically, whenever the naturally in any analysis that respects the affine system has any such. It means that the method structure and does not depend on the dimen- preserves affine symmetries, again automatically; sion. This explains their ubiquity. It is fascinating the method does not even have to ‘know’ (or be that natural and practical demands of numerical told) that the system has the symmetries. It means methods for ODE — black-box solvers defined that the method leaves decoupled systems decou- uniformly on all affine spaces — has led to the pled, again automatically. All these properties are discovery of a fundamental invariant object. desirable when designing general-purpose ODE On the other hand, where does this leave software. Furthermore, we now see that many the aromatic series? We suggest that they will of the more subtle properties of B-series, origi- show up naturally in problems posed in a specific nally discovered through combinatorial analysis dimension. Although traces and divergences are of trees, must in fact be a direct consequence of common in physics, we have not seen aromatic affine-relatedness. Examples include special prop- series before. They arose purely from a question erties with respect to symplecticity, preservation of in numerical analysis, but are fundamental in their quadratic invariants, and preservation of energy own way. Moreover, they can have properties that [14] and non-preservation of volume [26]. no B-series can have. For example, many aromatic The proof of the theorem on affine equivariance series, but no B-series, are divergence free. [34] relies on some classical results in functional Acknowledgements analysis and invariant theory. First it is established that the Taylor series in f of an arbitrary map We thank John Butcher, Ernst Hairer, and Gerhard depends only on the derivatives of f , and that Wanner for their comments. the terms of order n are in fact a of R I McLachlan is supported by the Mars- degree n in f and its partial derivatives. Second, den Fund of the Royal Society of New Zealand. the invariant polynomials that are functions of f K Modin is supported by the Swedish Founda- tion for Strategic Research (ICA12-0052) and EU and its partial derivatives, whose values at x0 are regarded now as arbitrary symmetric tensors, are Horizon 2020 Marie Sklodowska-Curie Individual sought using the ‘invariant tensor theorem’. The Fellowship (661482). i j i j conclusion at 2nd order is that only f fi and f fj are References equivariant, these giving the two aromatic trees of i j k [1] G. Bogfjellmo and A. Schmeding, The Lie group order 2. At 3rd order, to the tensor f f f the partial structure of the Butcher group, Found. Comp. Math. derivatives j and k can be attached to any two (2015), DOI:10.1007/s10208-015-9285-5

December 2017, Volume 7 No 1 9 Asia Pacific Mathematics Newsletter 10

[2] C. Brouder, Runge–Kutta methods and renormal- [21] E. Hairer, C. Lubich and G. Wanner, Geometric ization, Eur. Phys. J. C 12 (2000) 521–534. numerical integration: Structure-preserving algorithms [3] http://jcbutcher.com/publications for ordinary differential equations, 2nd ed., Springer, [4] J. C. Butcher, personal communication. Berlin, 2006. [5] J. C. Butcher, Coefficients for the study of Runge- [22] E. Hairer and G. Wanner, On the Butcher group Kutta integration processes, J. Austral. Math. Soc. 3 and general multi-value methods, Computing 13 (1963) 185–201. (1974) 1–15. [6] J. C. Butcher, Implicit Runge-Kutta processes, [23] E. Hairer, S. P. Nørsett and G. Wanner, Solving Math. Comp. 18 (1964) 50–64. ordinary differential equation I: Nonstiff problems, [7] J. C. Butcher, An algebraic theory of integration Springer, Berlin, 1987. methods, Math. Comp. 26 (1972) 79–106. [24] F. Harary, Independent discoveries in graph the- [8] J. C. Butcher, Numerical methods for ordinary ory, Ann. New York Acad. Sci. 328 (1979) 1–4. differential equations: early days, in The Birth of [25] A. Iserles, A first course in the numerical analysis of Numerical Analysis, A. Bultheel and R. Cools, eds., differential equations, Cambridge University Press, World Scientific, 2010, pp. 35–44. Cambridge, 2009. [9] J. C. Butcher and G. Wanner, Runge-Kutta meth- [26] A. Iserles, G. R. W. Quispel and P. S. P. Tse, ods: Some historical notes, Appl. Numer. Math. 22 B-series methods cannot be volume-preserving, (1996) 113–151. BIT Numerical Mathematics 47 (2007) 351–378. [10] D. Calaque, K. Ebrahimi-Fard and D. Manchon, [27] G. P. Jelliss, Knight’s Tour notes, http:// Two interacting Hopf algebras of trees: A Hopf- www.mayhematics.com/t/2n.htm. algebraic approach to composition and substi- [28] D. King-Hele, A Tapestry of Orbits, Cambridge tution of B-series, Adv. Appl. Math. 47 (2011) University Press, 2005. 282–308. [29] D. King-Hele, The destruction of the Royal Air- [11] P. Cartier, Vinberg algebras, Lie groups and com- craft Establishment, https://www.youtube.com/ binatorics, in Clay Mathematics Proceedings. Quanta watch?v=E0fSLiAa9Zw. of Maths 11 (2010) 107–126. [30] S. Khashin, Butcher algebras for Butcher systems, [12] A. Cayley, On the theory of the analytical forms Numer. Alg. 63 (2013) 679–689. called trees, Philos. Mag. 13(85) (1857) 172–176. [31] R. I. McLachlan, K. Modin, H. Munthe-Kaas [13] F. Chapoton and M. Livernet, Pre-Lie algebras and and O. Verdier, B-series are exactly the affine- the rooted trees operad, International Mathematics equivariant methods, Numer. Math. (2015), Research Notices 8 (2001) 395–408. pp. 1–24. [14] P. Chartier, E. Faou and M. Murua, An alge- [32] R. H. Merson, An operational method for the braic approach to invariant preserving integrators: study of integration processes, in Proceedings The case of quadratic and Hamiltonian invariants, of Conference on Data Processing and Automatic Numer. Math. 103 (2006) 575–590. Computing Machines vol. 1, Weapons Research [15] E. Celledoni, R. I. McLachlan, B. Owren and G. R. Establishment, Salisbury, South Australia, 1957, W. Quispel, Energy-preserving integrators and the pp. 1–25. structure of B-series, Foundations of Computational [33] H. Z. Munthe-Kaas and A. Lundervold, On Mathematics 10 (2010) 673–693. post-Lie algebras, Lie–Butcher series and moving [16] A. Connes and D. Kreimer, Lessons from quan- frames. Found. Comp. Math. 13 (2013) 583–613. tum field theory: Hopf algebras and spacetime [34] H. Munthe-Kaas and O. Verdier (2015). Aro- geometries. Letters in Mathematical Physics 48 (1999) matic Butcher series. Found. Comp. Math. 16 (2016) 85–96. 183–215. [17] A. Dur,¨ M¨obius functions, incidence algebras and [35] J. M. Sanz-Serna and A. Murua, Formal series power series representations, Springer, Berlin 1986, and numerical integrators: Some history and some pp. 88–90. new techniques, in Proceedings of the 8th Interna- [18] F. Fauvet, F. Menous and D. Sauzin, Explicit lin- tional Congress on Industrial and Applied Mathematics earization of one-dimensional germs through tree- (ICIAM 2015), Lei Guo and Zhi-Ming eds., Higher expansions, preprint, 2014. Education Press, Beijing, 2015, 311–331. [19] M. Gerstenhaber, The cohomology structure of an [36] E. B. Vinberg, The theory of convex homogeneous associative ring, Ann. Math. 78 (1963) 267–288. cones, Trans. Moscow Math. Soc. 12 (1963) 340–403. [20] M. Gubinelli, Ramification of rough paths, Journal [37] G. Wanner, personal communication. of Differential Equations 248 (2010) 693–721.

10 December 2017, Volume 7 No 1 Asia Pacific Mathematics Newsletter

Hans Munthe-Kaas University of Bergen, Norway [email protected]

Hans Munthe-Kaas is professor of Mathematics at the University of Bergen. He obtained his PhD from the Norwegian University of Science and Technology in 1989. In 1991 he joined the University of Bergen as professor of Computer Science. Much of the work of Munthe-Kaas concerns structure preserv- ing algorithms for integration of differential equations, and applications of group theory in computational science. He is Managing Editor of the journal 'Foundations of Computational Mathematics' and he is elected member of

Photo credit: Peter Badge the Norwegian Academy of Science and Letters. From 2018 to 2022, he will be chairing the international committee selecting the annual winner of the Abel prize in Mathematics.

Klas Modin Chalmers University of Technology and University of Gothenburg, Sweden

Klas Modin is a mathematician and Associate Professor at Chalmers University of Technology and University of Gothenburg in Sweden. He obtained his PhD from in 2010. After that he did two post-docs, first at Massey University in New Zealand, and then at the University of Toronto in Canada. He was recruited to Chalmers in 2013. Modin’s research interests lies in the borderland between geometry and computational mathematics.

Photo credit: Aldara Naveira Suarez

Robert McLachlan Massey University, New Zealand

Robert McLachlan FRSNZ is Distinguished Professor in Applied Mathematics at Massey University, New Zealand. He studied at Caltech with Herb Keller and worked at the University of Colorado, Boulder, and ETH Zurich, before joining Massey University in 1994. He is interested in applications of geometry to all areas of applied mathematics, including geometric numerical integra- tion, PDEs, dynamical systems, and image processing.

Olivier Verdier Western Norway University of Applied Sciences, Norway

Olivier Verdier is associate professor of mathematics in the Information and Communication Technology group in the Department of Computing, Mathematics and Physics of the Western Norway University of Applied Sciences. He obtained his PhD from Lund University in 2009. His main interests are Bayesian statistics and machine learning; functional program- ming, certified programming and type theory; and scientific computing and geometric integration.

December 2017, Volume 7 No 1 11