A theoretical framework for backward error analysis on manifolds

Backward Error Analysis (BEA) has been a crucial tool when analyzing long-time behavior of numerical integrators, in particular, one is interested in the geometric properties of the perturbed vector field that a numerical integrator generates. In this article we present a new framework for BEA on manifolds. We extend the previously known "exponentially close" estimates from $\mathbb{R}^n$ to smooth manifolds and also provide an abstract theory for classifications of numerical integrators in terms of their geometric properties. Classification theorems of type "symplectic integrators generate symplectic perturbed vector fields" are known to be true in $\mathbb{R}^n.$ We present a general theory for proving such theorems on manifolds by looking at the preservation of smooth $k$-forms on manifolds by the pull-back of a numerical integrator. This theory is related to classification theory of subgroups of diffeomorphisms. We also look at other subsets of diffeomorphisms that occur in the classification theory of numerical integrators. Typically these subsets are anti-fixed points of group homomorphisms.

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