Thesis Defense: Functional Equivariance and Backward Error Analysis

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.

 

Advisor: Ari Stern