43 research outputs found
Numerical wave optics and the lensing of gravitational waves by globular clusters
We consider the possible effects of gravitational lensing by globular
clusters on gravitational waves from asymmetric neutron stars in our galaxy. In
the lensing of gravitational waves, the long wavelength, compared with the
usual case of optical lensing, can lead to the geometrical optics approximation
being invalid, in which case a wave optical solution is necessary. In general,
wave optical solutions can only be obtained numerically. We describe a
computational method that is particularly well suited to numerical wave optics.
This method enables us to compare the properties of several lens models for
globular clusters without ever calling upon the geometrical optics
approximation, though that approximation would sometimes have been valid.
Finally, we estimate the probability that lensing by a globular cluster will
significantly affect the detection, by ground-based laser interferometer
detectors such as LIGO, of gravitational waves from an asymmetric neutron star
in our galaxy, finding that the probability is insignificantly small.Comment: To appear in: Proceedings of the Eleventh Marcel Grossmann Meetin
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Stability of Runge–Kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments
AbstractThis paper deals with the numerical properties of Runge–Kutta methods for the solution of u′(t)=au(t)+a0u([t+12]). It is shown that the Runge–Kutta method can preserve the convergence order. The necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained. It is interesting that the θ-methods with 0⩽θ<12 are asymptotically stable. Some numerical experiments are given
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances
Magnus integrators for solving linear-quadratic differential games
We consider Magnus integrators to solve linear-quadratic N-player differential games.
These problems require to solve, backward in time, non-autonomous matrix Riccati
differential equations which are coupled with the linear differential equations for the
dynamic state of the game, to be integrated forward in time. We analyze different
Magnus integrators which can provide either analytical or numerical approximations to the
equations. They can be considered as time-averaging methods and frequently are used as
exponential integrators. We show that they preserve some of the most relevant qualitative
properties of the solution for the matrix Riccati differential equations as well as for the
remaining equations. The analytical approximations allow us to study the problem in terms
of the parameters involved. Some numerical examples are also considered which show that
exponential methods are, in general, superior to standard methods.The authors acknowledge the support of the Generalitat Valenciana through the project GV/2009/032. The work of SB has also been partially supported by Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by the ERDF of the European Union) and the work of EP has also been partially supported by Ministerio de Ciencia e Innvacion of Spain, by the project MTM2009-08587.Blanes Zamora, S.; Ponsoda Miralles, E. (2012). Magnus integrators for solving linear-quadratic differential games. Journal of Computational and Applied Mathematics. 236(14):3394-3408. https://doi.org/10.1016/j.cam.2012.03.008S339434082361
A stroboscopic averaging algorithm for highly oscillatory delay problems
We propose and analyze a heterogenous multiscale method for the efficient
integration of constant-delay differential equations subject to fast periodic
forcing. The stroboscopic averaging method (SAM) suggested here may provide
approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a
computational effort that grows like \(H^{-1}\) (the inverse of the
stepsize), uniformly in the forcing frequency Omega
Ordnungssterne und Ordnungspfeile
Die Literatur von den Autoren Hairer, Wanner, Nørsett und Butcher, die der Arbeit als wichtige Quellen zugrunde lag, beschäftigt sich auch intensiv mit Mehrschrittverfahren.
Hier wird jeweils nur ein kurzer Ausblick auf Ordnungssterne bzw. Ordnungspfeile bei Mehrschrittverfahren mit einem Beispiel gegeben. Auch auf eine Behandlung der Ordnungssterne im Gebiet der Approximationstheorie wird mit einem Beispiel kurz hingewiesen