Bseries

B-series is a formal series used in the analysis of numerical integration methods for ordinary differential equations. It was introduced by John C. Butcher as a unifying framework for describing the effect of a numerical method on the solution of an ODE. The construction uses rooted trees, called elementary trees, to encode nested derivatives of the vector field.

For a vector field f on a manifold and a step size h, the B-series associated with a method has the form B(a, f, h)(x) = x + sum over rooted trees τ of aτ h^{|τ|} / s(τ) F(τ)(x), where F(τ) is the elementary differential corresponding to τ, |τ| is the number of nodes in τ, s(τ) is a symmetry factor, and aτ are real coefficients determined by the numerical method. The elementary differential F(τ) is defined recursively: F(•) = f, and F(τ) = f^{(k)}(x)[F(τ1), ..., F(τk)] when τ has a root connected to k subtrees τ1,...,τk.

B-series provide a compact, algebraic way to express and compare numerical methods, particularly Runge-Kutta and composition methods. They underpin the order conditions that determine the accuracy of a method and facilitate backward error analysis and the study of the Butcher group, an algebraic structure governing the composition of methods. Extensions exist for stochastic differential equations and other settings.

In practice, B-series enable researchers to analyze and design integrators with desired accuracy and structural properties, by translating methodic conditions into identities on rooted trees and their elementary differentials.