905 research outputs found
Multidomain Spectral Method for the Helically Reduced Wave Equation
We consider the 2+1 and 3+1 scalar wave equations reduced via a helical
Killing field, respectively referred to as the 2-dimensional and 3-dimensional
helically reduced wave equation (HRWE). The HRWE serves as the fundamental
model for the mixed-type PDE arising in the periodic standing wave (PSW)
approximation to binary inspiral. We present a method for solving the equation
based on domain decomposition and spectral approximation. Beyond describing
such a numerical method for solving strictly linear HRWE, we also present
results for a nonlinear scalar model of binary inspiral. The PSW approximation
has already been theoretically and numerically studied in the context of the
post-Minkowskian gravitational field, with numerical simulations carried out
via the "eigenspectral method." Despite its name, the eigenspectral technique
does feature a finite-difference component, and is lower-order accurate. We
intend to apply the numerical method described here to the theoretically
well-developed post-Minkowski PSW formalism with the twin goals of spectral
accuracy and the coordinate flexibility afforded by global spectral
interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes
revisions based on referee reports and has two extra figure
Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains
We describe a multidomain spectral-tau method for solving the
three-dimensional helically reduced wave equation on the type of two-center
domain that arises when modeling compact binary objects in astrophysical
applications. A global two-center domain may arise as the union of Cartesian
blocks, cylindrical shells, and inner and outer spherical shells. For each such
subdomain, our key objective is to realize certain (differential and
multiplication) physical-space operators as matrices acting on the
corresponding set of modal coefficients. We achieve sparse banded realizations
through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and
Torres. Since ours is the first three-dimensional multidomain implementation of
the technique, we focus on the issue of convergence for the global solver, here
the alternating Schwarz method accelerated by GMRES. Our methods may prove
relevant for numerical solution of other mixed-type or elliptic problems, and
in particular for the generation of initial data in general relativity.Comment: 37 pages, 3 figures, 12 table
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
Monte Carlo simulation of SU(2) Yang-Mills theory with light gluinos
In a numerical Monte Carlo simulation of SU(2) Yang-Mills theory with light
dynamical gluinos the low energy features of the dynamics as confinement and
bound state mass spectrum are investigated. The motivation is supersymmetry at
vanishing gluino mass. The performance of the applied two-step multi-bosonic
dynamical fermion algorithm is discussed.Comment: latex, 48 pages, 16 figures with epsfi
Automatic Deformation of Riemann-Hilbert Problems with Applications to the Painlev\'e II Transcendents
The stability and convergence rate of Olver's collocation method for the
numerical solution of Riemann-Hilbert problems (RHPs) is known to depend very
sensitively on the particular choice of contours used as data of the RHP. By
manually performing contour deformations that proved to be successful in the
asymptotic analysis of RHPs, such as the method of nonlinear steepest descent,
the numerical method can basically be preconditioned, making it asymptotically
stable. In this paper, however, we will show that most of these preconditioning
deformations, including lensing, can be addressed in an automatic, completely
algorithmic fashion that would turn the numerical method into a black-box
solver. To this end, the preconditioning of RHPs is recast as a discrete,
graph-based optimization problem: the deformed contours are obtained as a
system of shortest paths within a planar graph weighted by the relative
strength of the jump matrices. The algorithm is illustrated for the RHP
representing the Painlev\'e II transcendents.Comment: 20 pages, 16 figure
Spectral multigrid methods for elliptic equations 2
A detailed description of spectral multigrid methods is provided. This includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems. The spectral methods for periodic problems use Fourier series and those for Dirichlet problems are based upon Chebyshev polynomials. An improved preconditioning for Dirichlet problems is given. Numerical examples and practical advice are included
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