Hyperbolic Trajectories of Time Discretizations
نویسنده
چکیده
A new paradigm for numerically approximating trajectories of an ODE is espoused. We ask for a one to one correspondence between trajectories of an ODE and its discrete approximation. The results enable one, in principal, to compute a trajectory of a discrete approximation, and to use this computation to rigorously prove the existence of a trajectory of the ODE near the discrete trajectory. More precisely, we formulate an appropriate notion of hyperbolicity for a bounded trajectory of a discrete approximation to an ODE. Our de nition is motivated from the continuous case. An example of a discrete trajectory satisfying our de nition of hyperbolicity is given. We show that the underlying ODE inherits a unique nearby trajectory which is hyperbolic in the continuous sense. Periodicity is also inherited. The discrete trajectory converges to the continuous trajectory with the `correct' order of convergence. The strength of the hyperbolicity may tend to zero as the accuracy of the approximation increases, and our result will still hold. This is quanti ed in a precise way. Explicit estimates of all pertinent quantities for applying the result in practice are given. In the course of the proof, we state a result (proved elsewhere) which gives precise estimates of the spectra of perturbed linear di erence equations, which may be of independent interest.
منابع مشابه
AFRL-OSR-VA-TR-2015-0114 Tailoring high order time discretizations for use with spatal discretizations of hyperbolic PDEs
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