Abstract

Geometric, or structure-preserving, numerical integration has long been used as a framework for studying integrators that preserve a systems invariants. In backward error analysis, the approximate numerical flow of a system is viewed as the exact flow of a modified problem, allowing us to gain qualitative insights into the behavior of the numerical solution. The preservation of conservation laws by a numerical integrator can be generalized to $F$-functionally equivariant integrators, where $F$ represents an observable of the system in consideration. This thesis describes the behavior of geometric integrators through the lens of backward error analysis. First, we extend the idea of $F$-functional equivariance to modified vector fields, generalizing results on invariant preservation and describing the numerical evolution of non-invariant observables. Next, we introduce algebraic conditions for $F$-functionally equivariant B-series methods and their modified vector fields. A special case of this condition, when $F$ is quadratic, yields the well-known algebraic characterization of symplectic B-series. Finally, we define conjugate functionally equivariant integrators and develop the notion of modified observables with respect to such integrators, analogous to modified quadratic observables, in the context of near symplectic integrators.

Committee Chair

Ari Stern

Committee Members

Ari Stern; Donsub Rim; Gilles Vilmart; Maike Sonnewald; Robert McLachlan; Xiang Tang

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

7-3-2025

Language

English (en)

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Available for download on Thursday, July 02, 2026

Included in

Mathematics Commons

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