6,619 research outputs found
Filtering in Legendre spectral methods
We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments
A Statistical Toolbox For Mining And Modeling Spatial Data
Most data mining projects in spatial economics start with an evaluation of a set of attribute variables on a sample of spatial entities, looking for the existence and strength of spatial autocorrelation, based on the Moran’s and the Geary’s coefficients, the adequacy of which is rarely challenged, despite the fact that when reporting on their properties, many users seem likely to make mistakes and to foster confusion. My paper begins by a critical appraisal of the classical definition and rational of these indices. I argue that while intuitively founded, they are plagued by an inconsistency in their conception. Then, I propose a principled small change leading to corrected spatial autocorrelation coefficients, which strongly simplifies their relationship, and opens the way to an augmented toolbox of statistical methods of dimension reduction and data visualization, also useful for modeling purposes. A second section presents a formal framework, adapted from recent work in statistical learning, which gives theoretical support to our definition of corrected spatial autocorrelation coefficients. More specifically, the multivariate data mining methods presented here, are easily implementable on the existing (free) software, yield methods useful to exploit the proposed corrections in spatial data analysis practice, and, from a mathematical point of view, whose asymptotic behavior, already studied in a series of papers by Belkin & Niyogi, suggests that they own qualities of robustness and a limited sensitivity to the Modifiable Areal Unit Problem (MAUP), valuable in exploratory spatial data analysis
Large-eddy simulation of the lid-driven cubic cavity flow by the spectral element method
This paper presents the large-eddy simulation of the lid-driven cubic cavity
flow by the spectral element method (SEM) using the dynamic model. Two spectral
filtering techniques suitable for these simulations have been implemented.
Numerical results for Reynolds number are showing very good
agreement with other experimental and DNS results found in the literature
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
A coupled approximate deconvolution and dynamic mixed scale model for large-eddy simulation
Large-eddy simulations of incompressible Newtonian fluid flows with
approximate deconvolution models based on the van Cittert method are reported.
The Legendre spectral element method is used for the spatial discretization to
solve the filtered Navier--Stokes equations. A novel variant of approximate
deconvolution models blended with a mixed scale model using a dynamic
evaluation of the subgrid-viscosity constant is proposed. This model is
validated by comparing the large-eddy simulation with the direct numerical
simulation of the flow in a lid-driven cubical cavity, performed at a Reynolds
number of 12'000. Subgrid modeling in the case of a flow with coexisting
laminar, transitional and turbulent zones such as the lid-driven cubical cavity
flow represents a challenging problem. Moreover, the coupling with the spectral
element method having very low numerical dissipation and dispersion builds a
well suited framework to analyze the efficiency of a subgrid model. First- and
second-order statistics obtained using this new model are showing very good
agreement with the direct numerical simulation. Filtering operations rely on an
invertible filter applied in a modal basis and preserving the C0-continuity
across elements. No clipping on dynamic parameters was needed to preserve
numerical stability
A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations
Recent work by McClarren & Hauck [29] suggests that the filtered spherical
harmonics method represents an efficient, robust, and accurate method for
radiation transport, at least in the two-dimensional (2D) case. We extend their
work to the three-dimensional (3D) case and find that all of the advantages of
the filtering approach identified in 2D are present also in the 3D case. We
reformulate the filter operation in a way that is independent of the timestep
and of the spatial discretization. We also explore different second- and
fourth-order filters and find that the second-order ones yield significantly
better results. Overall, our findings suggest that the filtered spherical
harmonics approach represents a very promising method for 3D radiation
transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of
Computational Physic
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