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This work is concerned with the global continuation for solutions (λ,u,ξ) ∈ R × C1{0}([0,∞), RN) × Rk of the following system of ordinary differential equations: where F: [0,∞) × RN × U × J → RN and φ: U × J → X1, for some open sets J ⊂ R and U ⊂ Rk, and where RN = X1 ⊕ X2 is a given decomposition, with associated projection P: RN → X1. This problem gives rise to a nonlinear operator whose zeros correspond exactly to the solutions of the original problem. We give conditions on F and φ ensuring that this operator has the Fredholm property and is proper on the closed bounded subsets of R × C1{0}([0,∞), RN) × Rk. These conditions generalize to a parameter dependent situation some recent results obtained by Morris [24]. Under these assumptions, we study the global behavior of a particular connected set of solutions using a degree theory available for such operators and obtain global continuation theorems. In the second part of this work, we use our general results to prove the existence of solutions for the so-called swirling flow problem in fluid dynamics, which can be written as a system of two ordinary differential equations on the half-line together with boundary conditions. Having obtained a priori bounds on possible solutions to this problem, we are able to recover the existence result obtained by Mcleod [22] using ad hoc arguments. In the last part, we present some numerical computations and give pictures of solutions to this problem.
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how?' but also
why?', where?' and
what for?'.
The motivation for developing structure-preserving algorithms for special classes of problems originates independently in such diverse areas of research as astronomy, molecular dynamics, mechanics, control theory, theoretical physics and numerical analysis, with important contributions from other areas of both applied and pure mathematics. Moreover, it turns out that preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods.
In addition to the construction of geometric integrators, an important aspect of geometric integration is the light it sheds on the relationship between geometric properties of a numerical method and favourable error propagation in long-time integration. A very successful organising principle is backward error analysis, whereby the numerical one-step map is interpreted as (almost) the flow of a modified differential equation. In this way, geometric properties of the numerical integrator translate seamlessly into structure preservation on the level of the modified equation. Much insight and rigourous error estimates over long time intervals can then be obtained by combining backward error analysis with the KAM theory and related perturbation theories for Hamiltonian and reversible systems. While this approach has been very successful for ordinary differential equations, much less is currently known about highly oscillatory systems and geometric integrators for partial differential equations.
Geometric numerical integration has been an active interdisciplinary research area since the last decade. Although the subject is in a lively phase of intensive development, the results so far are substantive and they impact on a wide range of application areas and on our understanding of core issues in computational mathematics. This is evidenced by the monographs \cite{HLW:GNI2002,LR:SMH2004}.