12,693 research outputs found
On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation
In this paper we develop further a method for detecting unstable periodic
orbits (UPOs) by stabilising transformations, where the strategy is to
transform the system of interest in such a way that the orbits become stable.
The main difficulty of using this method is that the number of transformations,
which were used in the past, becomes overwhelming as we move to higher
dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have
recently proposed a set of stabilising transformations which is constructed
from a small set of already found UPOs (Crofts and Davidchack 2006). The main
benefit of using the proposed set is that its cardinality depends on the
dimension of the unstable manifold at the UPO rather than the dimension of the
system. In a typical situation the dimension of the unstable manifold is much
smaller than the dimension of the system so the number of transformations is
much smaller. Here we extend this approach to high-dimensional systems of ODEs
and apply it to the model example of a chaotic spatially extended system -- the
Kuramoto-Sivashinsky equation. A comparison is made between the performance of
this new method against the competing methods of Newton-Armijo (NA) and
Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact
that the LM algorithm is able to solve under-determined systems of equations,
thus eliminating the need for any additional constraints
The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method
The periodic standing wave (PSW) method for the binary inspiral of black
holes and neutron stars computes exact numerical solutions for periodic
standing wave spacetimes and then extracts approximate solutions of the
physical problem, with outgoing waves. The method requires solution of a
boundary value problem with a mixed (hyperbolic and elliptic) character.
We present here a new numerical method for such problems, based on three
innovations: (i) a coordinate system adapted to the geometry of the problem,
(ii) an expansion in multipole moments of these coordinates and a filtering out
of higher moments, and (iii) the replacement of the continuum multipole moments
with their analogs for a discrete grid. We illustrate the efficiency and
accuracy of this method with nonlinear scalar model problems. Finally, we take
advantage of the ability of this method to handle highly nonlinear models to
demonstrate that the outgoing approximations extracted from the standing wave
solutions are highly accurate even in the presence of strong nonlinearities.Comment: RevTex, 32 pages, 13 figures, 6 table
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator
This paper introduces a novel method for the efficient and accurate
computation of the volume of a domain whose boundary is given by an orientable
hypersurface which is implicitly given as the iso-contour of a sufficiently
smooth level-set function. After spatial discretization, local approximation of
the hypersurface and application of the Gaussian divergence theorem, the volume
integrals are transformed to surface integrals. Application of the surface
divergence theorem allows for a further reduction to line integrals which are
advantageous for numerical quadrature. We discuss the theoretical foundations
and provide details of the numerical algorithm. Finally, we present numerical
results for convex and non-convex hypersurfaces embedded in cuboidal domains,
showing both high accuracy and thrid- to fourth-order convergence in space.Comment: 25 pages, 17 figures, 3 table
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