Multi-indice B-series

TL;DR

This paper introduces multi-indices as a new combinatorial tool to analyze numerical methods for ordinary differential equations, offering a more compact representation than traditional rooted trees in B-series.

Contribution

It replaces rooted trees with multi-indices in B-series, enabling a compressed description of numerical schemes and unique characterization of Taylor expansions for certain maps.

Findings

Multi-indices provide a more compact B-series representation.

Multi-indices uniquely characterize Taylor expansions of local maps.

The approach bridges numerical analysis and stochastic PDE solution descriptions.

Abstract

We propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's B-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi-indices B-series uniquely characterize the Taylor expansion of one-dimensional local and affine equivariant maps.