Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-indice B-series

Yvain Bruned, Kurusch Ebrahimi-Fard, Yingtong Hou

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.

Multi-indice B-series

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.
Paper Structure (5 sections, 21 theorems, 91 equations, 1 algorithm)

This paper contains 5 sections, 21 theorems, 91 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multi-indices
  3. Composition of multi-indice $B$-series
  4. Substitution of multi-indice $B$-series
  5. Connection with local affine equivariant methods

Key Result

Proposition 1.2

Let $(\mathfrak g, \triangleright)$ be a left pre-Lie algebra. The commutator bracket defines a Lie algebra on $\mathfrak g$, i.e., it satisfies the Jacobi identity for all $x, y \in \mathfrak g$.

Theorems & Definitions (38)

  • Definition 1.1
  • Proposition 1.2
  • Example 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.4
  • Corollary 2.5
  • Remark 2.6
  • Example 2.7
  • ...and 28 more