Let be an analytic function satisfying an ordinary differential equation . A Butcher series for is a series of the form
where is the set of rooted trees, is the coefficient for the series associated with the tree , is the elementary differential associated with the tree , is a stepsize or -increment, and , often called the order of the tree , is the number of nodes in the tree . All terms will be defined below. For a more thorough discussion of Butcher series, see [a1] or [a2].
A tree is a connected acyclic graph (cf. also Graph). A rooted tree is a tree with a distinguished node, called the root (which is drawn at the bottom of the sample trees in the Table). Two rooted trees and are isomorphic (i.e., equivalent) if and only if there is a bijection from the nodes of to the nodes of such that maps the root of to the root of and any two nodes and are connected by an edge in if and only if and are connected by an edge in .
The following bracket notation for rooted trees is very useful. Let and be the trees with zero and one node, respectively. For rooted trees with for , let be the rooted tree formed by connecting a new root to the roots of each of . Clearly, and any rooted tree formed by permuting the subtrees of is isomorphic to . The rooted trees with at most four nodes are displayed in the table below.
Using this bracket notation for rooted trees, one can recursively define elementary differentials, as follows. Let
and, for with for ,
where , the th derivative of with respect to , is a multi-linear mapping.
Since satisfies , it is not hard to show that
where is the th derivative of with respect to , , and is the number of distinct ways of labeling the nodes of with the integers such that the labels increase as you follow any path from the root to a leaf of . Consequently, one can write the Taylor series for as a Butcher series:
The importance of Butcher series stems from their use to derive and to analyze numerical methods for differential equations. For example, consider the -stage Runge–Kutta formula (cf. Runge–Kutta method)
for the ordinary differential equation . Let be the vector of weights and the coefficient matrix for (a2). Then it can be shown that can be written as the Butcher series
where the coefficients , defined below, depend only on the rooted tree and the coefficients of the formula (a2).
To define , first let
and then, for with for , define recursively by
where the product of 's in the bracket is taken component-wise. Then
and, for with for ,
where, again, the product of 's in the bracket is taken componentwise.
Assuming that and that for all trees of order , it follows immediately from (a1) and (a3) that
The order of (a2) is the largest such .
|[a1]||J.C. Butcher, "The numerical analysis of ordinary differential equations" , Wiley (1987)|
|[a2]||E. Hairer, S.P. Nørsett, G. Wanner, "Solving ordinary differential equations I: nonstiff problems" , Springer (1987)|
Butcher series. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Butcher_series&oldid=35592