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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 11 Dec 2023 12:51:47 +0000Stable Attracting Sets in Dynamical Systems and in Their One-Step Discretizations
https://resolver.caltech.edu/CaltechAUTHORS:20120627-134902327
Authors: Kloeden, P. E.; Lorenz, J.
Year: 1986
DOI: 10.1137/0723066
We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape. Under the assumption of uniform asymptotic stability of Λ in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of Λ.https://authors.library.caltech.edu/records/ynb5n-q0564On the Rate of Convergence of Viscosity Solutions for Boundary Value Problems
https://resolver.caltech.edu/CaltechAUTHORS:LORsiamjma87
Authors: Lorenz, Jens; Sanders, Richard
Year: 1987
DOI: 10.1137/0518024
A class of singularly perturbed boundary value problems is considered for viscosity tending to zero. From compactness arguments it is known that the solutions converge to a limit function characterized by an entropy inequality. We formulate an approximate entropy inequality (AEI) and use it to obtain the order of convergence. The AEI is also used to obtain the order of convergence for monotone difference schemes.https://authors.library.caltech.edu/records/sr4nq-2xr38A High-Order Method for Stiff Boundary Value Problems with Turning Points
https://resolver.caltech.edu/CaltechAUTHORS:20120626-164218625
Authors: Brown, David L.; Lorenz, Jens
Year: 1987
DOI: 10.1137/0908067
This paper describes some high-order collocation-like methods for the numerical solution of stiff boundary-value problems with turning points. The presentation concentrates on the implementation of these methods in conjunction with the implementation of the a priori mesh construction algorithm introduced by Kreiss, Nichols and Brown [SIAM J. Numer. Anal., 23 (1986), pp. 325–368] for such problems. Numerical examples are given showing the high accuracy which can be obtained in solving the boundary value problem for singularly perturbed ordinary differential equations with turning points.https://authors.library.caltech.edu/records/d6vgs-z6v37Applications of Adaptive Data Distributions
https://resolver.caltech.edu/CaltechAUTHORS:20170627-165132372
Authors: Van de Velde, Eric F.; Lorenz, Jens
Year: 1990
DOI: 10.1109/DMCC.1990.555391
Continuation methods compute paths of solutions of nonlinear equations that depend on a parameter. This paper examines some aspects of the multicomputer implementation of such methods. The computation is done on the Symult Series 2010 multicomputer.
One of the main issues in the development of concurrent programs is load balancing, achieved here by using appropriate data distributions. In the continuation process, a large number of linear systems have to be solved. For nearby points along the solution path, the corresponding system matrices are closely related to each other. Therefore, pivots which are good for the LU-decomposition of one matrix are likely to be acceptable for a whole segment of the solution path. This suggests to choose certain data distributions that achieve good load balancing. In addition, if these distributions are used, the resulting code is easily vectorized.
To test this technique, the invariant manifold of a system of two identical nonlinear oscillators is computed as a function of the coupling between them. This invariant manifold is determined by the solution of a system of nonlinear partial differential equations that depends on the coupling parameter. A symmetry in the problem reduces this system to one single equation, which is discretized by finite differences. The solution of this discrete nonlinear system is followed as the coupling parameter is changed.https://authors.library.caltech.edu/records/ngken-yk115