Core Concepts
Replace rooted trees with multi-indices in Butcher's B-series for numerical methods.
Abstract
The content introduces multi-indices as a novel way to describe numerical methods for ordinary differential equations, replacing rooted trees in Butcher's B-series. The composition and substitution of multi-indices B-series are explored, highlighting their connection with local affine equivariant methods. The significance of multi-indices in numerical analysis and singular SPDEs is emphasized.
- Introduction to classical B-series by Butcher.
- Multi-indices definition and application in numerical methods.
- Composition and substitution of multi-indices B-series.
- Connection with local affine equivariant methods.
- Importance of multi-indices in numerical analysis and SPDEs.
Stats
"Classical B-series play a pivotal role in the analysis of numerical integrators."
"Cayley discovered a correspondence between non-planar rooted trees and vector fields."
"The commutator bracket defines a Lie algebra on g satisfying the Jacobi identity."
Quotes
"Numerical methods that can be expanded in B-series correspond to sequences of maps—one map for each dimension." - McLachlan et al.
"Aromatic B-series methods characterize all local and affine equivariant methods." - Munthe-Kaas and Verdier