Asia Pacific Mathematics 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, polynomials 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.