Section: New Results

An algebraic theory of order

Participant : Philippe Chartier.

This a joint work with Ander Murua, from the University of the Basque Country.

When one needs to compute the numerical solution of a differential equation of a specific type (ordinary, differential-algebraic, linear...) with a method of a given class of numerical schemes, a deciding criterion to pick up the right one is its order of convergence: the systematic determination of order conditions thus appears as a pivotal question in the numerical analysis of differential equations. Given a family of vector fields with some specific property (say for instance linear, additively split into a linear and a nonlinear part, scalar...) and a set of numerical schemes (rational approximations of the exponential, exponential integrators , Runge-Kutta methods...), a fairly general recipe consists in expanding into series both the exact solution of the problem and its numerical approximation: order conditions are then derived by comparing the two series term by term, once their independence has been established. Depending on the equation and on the numerical method, these series can be indexed by integers or trees, and can be expressed in terms of elementary differentials or commutators of Lie-operators. Despite the great variety of situations encountered in practice and of ad-hoc techniques, the problems raised are strikingly similar and can be described as follows:

• is it possible to construct a set of algebraically independent order conditions?

• what are the order conditions corresponding to a scheme obtained by composition of two given methods?

• are there numerical schemes within the class considered of arbitrarily high order for arbitrary vector field?

• are there numerical schemes within the set of methods considered that approximate modified fields?

The Butcher group [35] and its underlying Hopf algebra of rooted trees were originally formulated to address these questions for Runge-Kutta methods. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry by Connes and Moscovici [38] and they describe the combinatorics of renormalization in quantum field theory as described by Kreimer [48] . In the present work, we show that the Hopf algebra of rooted trees associated to Butcher's group can be seen as a particular instance of a more general construction: given a group G of integrations schemes (satisfying some natural assumptions), we exhibit a sub-algebra of the algebra of functions acting on G , which is graded, commutative and turns out to be a Hopf algebra. Within this algebraic framework, we then address the questions listed above and provide answers that are relevant to many practical situations.

The paper [16] introduces an algebraic concept, called group of asbtract integration schemes , composed of a group of integrators G , an algebra H of functions on G and a scaling map whose existence is essential to the subsequent results. We begin by proving that, under some reasonable assumptions of a purely algebraic nature, the algebra H can be equipped with a co-product, an antipode and an embedded family of equivalence relations (called order), thus giving rise to a graded Hopf algebra structure. In particular, the co-product of H is per se the key to the second question in our list. It furthermore endows the linear dual H^* of H with an algebra structure, where a new group and a Lie-algebra can be defined and related through the exponential and logarithm maps. These two structures are of prime interest, since can be interpreted, in the more usual terminology of ODEs, as the set of “modified vector fields", while can be intrepreted as the larger group of “integrators” containing G . We then prove that all elements of can be “approximated” up to any order by elements of G . Although there seems to be no appropriate topology for G and , we can interpret this result by saying that G is dense in : this anwers the third and fourth questions in our list. Note that the proof of this result also provides a positive answer to the first question of our list.

Finally, we describe how our theory can be used to obtain order conditions for composition schemes.