20,034 research outputs found
On the efficiency and accuracy of interpolation methods for spectral codes
In this paper a general theory for interpolation methods on a rectangular
grid is introduced. By the use of this theory an efficient B-spline based
interpolation method for spectral codes is presented. The theory links the
order of the interpolation method with its spectral properties. In this way
many properties like order of continuity, order of convergence and magnitude of
errors can be explained. Furthermore, a fast implementation of the
interpolation methods is given. We show that the B-spline based interpolation
method has several advantages compared to other methods. First, the order of
continuity of the interpolated field is higher than for other methods. Second,
only one FFT is needed whereas e.g. Hermite interpolation needs multiple FFTs
for computing the derivatives. Third, the interpolation error almost matches
the one of Hermite interpolation, a property not reached by other methods
investigated.Comment: 19 pages, 5 figure
Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
We describe a modification of a fourth-order accurate ``moving puncture''
evolution code, where by replacing spatial fourth-order accurate differencing
operators in the bulk of the grid by a specific choice of sixth-order accurate
stencils we gain significant improvements in accuracy. We illustrate the
performance of the modified algorithm with an equal-mass simulation covering
nine orbits.Comment: 13 pages, 6 figure
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
Distributed-memory large deformation diffeomorphic 3D image registration
We present a parallel distributed-memory algorithm for large deformation
diffeomorphic registration of volumetric images that produces large isochoric
deformations (locally volume preserving). Image registration is a key
technology in medical image analysis. Our algorithm uses a partial differential
equation constrained optimal control formulation. Finding the optimal
deformation map requires the solution of a highly nonlinear problem that
involves pseudo-differential operators, biharmonic operators, and pure
advection operators both forward and back- ward in time. A key issue is the
time to solution, which poses the demand for efficient optimization methods as
well as an effective utilization of high performance computing resources. To
address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov
solver. Our algorithm integrates several components: a spectral discretization
in space, a semi-Lagrangian formulation in time, analytic adjoints, different
regularization functionals (including volume-preserving ones), a spectral
preconditioner, a highly optimized distributed Fast Fourier Transform, and a
cubic interpolation scheme for the semi-Lagrangian time-stepping. We
demonstrate the scalability of our algorithm on images with resolution of up to
on the "Maverick" and "Stampede" systems at the Texas Advanced
Computing Center (TACC). The critical problem in the medical imaging
application domain is strong scaling, that is, solving registration problems of
a moderate size of ---a typical resolution for medical images. We are
able to solve the registration problem for images of this size in less than
five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA;
November 201
Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows
We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable
High accuracy binary black hole simulations with an extended wave zone
We present results from a new code for binary black hole evolutions using the
moving-puncture approach, implementing finite differences in generalised
coordinates, and allowing the spacetime to be covered with multiple
communicating non-singular coordinate patches. Here we consider a regular
Cartesian near zone, with adapted spherical grids covering the wave zone. The
efficiencies resulting from the use of adapted coordinates allow us to maintain
sufficient grid resolution to an artificial outer boundary location which is
causally disconnected from the measurement. For the well-studied test-case of
the inspiral of an equal-mass non-spinning binary (evolved for more than 8
orbits before merger), we determine the phase and amplitude to numerical
accuracies better than 0.010% and 0.090% during inspiral, respectively, and
0.003% and 0.153% during merger. The waveforms, including the resolved higher
harmonics, are convergent and can be consistently extrapolated to
throughout the simulation, including the merger and ringdown. Ringdown
frequencies for these modes (to ) match perturbative
calculations to within 0.01%, providing a strong confirmation that the remnant
settles to a Kerr black hole with irreducible mass and spin $S_f/M_f^2 = 0.686923 \pm 10\times10^{-6}
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Quantitative Spectroscopy In 3D
The Advanced Spectrum Synthesis 3D Tool ASSET is introduced. ASSET allows for the accurate and fast calculation of spectra from 3D hydrodynamical models. To achieve the highest numerical accuracy 3(rd)-order Bezier interpolations are employed and all available information from the model grid with respect to the spacial and frequency resolution is exploited. ASSET is fully parallelized with OpenMP and MPI and highly optimized to run at about 25% of peak speed on workstations and clusters. The emergent flux for a single spectral line can be calculated from dozens of snapshots within a few minutes, and the whole spectrum (2 . 10(6) frequencies) can be calculated on a small cluster (a few hundred threads) within a day. The numerical methods, the serial optimization and the parallel implementation are described in some detail.Texas Advanced Computing Center (TACC
Spectral multigrid methods with applications to transonic potential flow
Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil
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