1,062 research outputs found
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
PAMELA: An Open-Source Software Package for Calculating Nonlocal Exact Exchange Effects on Electron Gases in Core-Shell Nanowires
We present a new pseudospectral approach for incorporating many-body,
nonlocal exact exchange interactions to understand the formation of electron
gases in core-shell nanowires. Our approach is efficiently implemented in the
open-source software package PAMELA (Pseudospectral Analysis Method with
Exchange & Local Approximations) that can calculate electronic energies,
densities, wavefunctions, and band-bending diagrams within a self-consistent
Schrodinger-Poisson formalism. The implementation of both local and nonlocal
electronic effects using pseudospectral methods is key to PAMELA's efficiency,
resulting in significantly reduced computational effort compared to
finite-element methods. In contrast to the new nonlocal exchange formalism
implemented in this work, we find that the simple, conventional
Schrodinger-Poisson approaches commonly used in the literature (1) considerably
overestimate the number of occupied electron levels, (2) overdelocalize
electrons in nanowires, and (3) significantly underestimate the relative energy
separation between electronic subbands. In addition, we perform several
calculations in the high-doping regime that show a critical tunneling depth
exists in these nanosystems where tunneling from the core-shell interface to
the nanowire edge becomes the dominant mechanism of electron gas formation.
Finally, in order to present a general-purpose set of tools that both
experimentalists and theorists can easily use to predict electron gas formation
in core-shell nanowires, we document and provide our efficient and
user-friendly PAMELA source code that is freely available at
http://alum.mit.edu/www/usagiComment: Accepted by AIP Advance
High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures
The work reported in this article presents a high-order, stable, and
efficient Gegenbauer pseudospectral method to solve numerically a wide variety
of mathematical models. The proposed numerical scheme exploits the stability
and the well-conditioning of the numerical integration operators to produce
well-conditioned systems of algebraic equations, which can be solved easily
using standard algebraic system solvers. The core of the work lies in the
derivation of novel and stable Gegenbauer quadratures based on the stable
barycentric representation of Lagrange interpolating polynomials and the
explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous
error and convergence analysis of the proposed quadratures is presented along
with a detailed set of pseudocodes for the established computational
algorithms. The proposed numerical scheme leads to a reduction in the
computational cost and time complexity required for computing the numerical
quadrature while sharing the same exponential order of accuracy achieved by
Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical
test examples to assess the efficiency and accuracy of the numerical scheme.
The present method provides a strong addition to the arsenal of numerical
pseudospectral methods, and can be extended to solve a wide range of problems
arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl
Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method
This paper presents a new effective radial basis function (RBF) collocation technique for the free vibration
analysis of laminated composite plates using the first order shear deformation theory (FSDT). The plates, which can be rectangular or non-rectangular, are simply discretised by means of Cartesian grids. Instead of using conventional differentiated RBF networks, one-dimensional integrated RBF networks (1D-IRBFN) are employed on grid lines to approximate the field variables. A number of examples concerning various thickness-to-span ratios, material properties and boundary conditions are considered. Results obtained are compared with the exact solutions and numerical results by other techniques in the literature to
investigate the performance of the proposed method
An effective spectral collocation method for the direct solution of high-order ODEs
This paper reports a new Chebyshev spectral collocation method for directly solving high-order ordinary differential equations (ODEs). The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows the multiple boundary conditions to be incorporated more efficiently. Numerical results show that the
proposed formulation significantly improves the conditioning of the system and yields more accurate results and faster convergence rates than conventional formulations
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