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31.
Summary. We examine the use of orthogonal spline collocation for the
semi-discreti\-za\-tion of the cubic Schr\"{o}dinger equation and the
two-dimensional
parabolic equation of Tappert. In each case, an optimal order
estimate of the error in the semidiscrete
approximation is derived. For the cubic Schr\"{o}dinger equation, we
present the results
of numerical experiments in which the integration in time is
performed using a routine from a software library.
Received February 14, 1992 / Revised version received December 29,
1992 相似文献
32.
33.
Explicit expressions for the eigensystems of one-dimensional finite element Galerkin (FEG) matrices based on C
0 piecewise quadratic polynomials are determined. These eigensystems are then used in the formulation of fast direct methods,
matrix decomposition algorithms (MDAs), for the solution of the FEG equations arising from the discretization of Poisson’s
equation on the unit square subject to several standard boundary conditions. The MDAs employ fast Fourier transforms and require
O(N
2log N) operations on an N×N uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms. 相似文献
34.
We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when
various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems
in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional
problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can
be employed in an MDA with a resulting computational cost of O(N
2 logN) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices
arising in corresponding two–point boundary value problems in one space dimension. In many important cases, these eigensystems
are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite
difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation,
finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily
on the Dirichlet problem for Poisson’s equation in the unit square, sketch extensions to other boundary conditions and to
more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems
in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising
from discretizations of non-separable boundary value problems. 相似文献
35.
Benjamin E. Blass Keith Coburn Neil Fairweather Andrew Fluxe Steve Hodson Chris Jackson John Janusz Wenlin Lee Jim Ridgeway Ron White Shengde Wu 《Tetrahedron letters》2006,47(42):7497-7499
An examination of the methods required for the amination of 2- and 4-imidazolidinones is described. 相似文献
36.
Orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the
unit square. Particular attention is given to the Dirichlet biharmonic problem which is solved using capacitance matrix techniques.
Received November 11, 1996 相似文献
37.
Graeme Fairweather Andreas Karageorghis Jon Maack 《Journal of computational physics》2011,230(8):2880-2895
Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N2 log N) on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition. 相似文献
38.
Particle dynamics in a channel flow are investigated using large eddy simulation and a Lagrangian particle tracking technique. Following validation of single-phase flow predictions against DNS results, fluid velocities are subsequently used to study the behaviour of particles of differing shape assuming one-way coupling between the fluid and the particles. The influence of shape- and orientation-dependent drag and lift forces on both the translational and rotational motion of the particles is accounted for to ensure accurate representation of the flow dynamics of non-spherical particles. The size of the particles studied was obtained based on an equivalent-volume sphere, and differing shapes were modelled using super-quadratic ellipsoid forms by varying their aspect ratio, with their orientation predicted using the incidence angle between the particle relative velocity and the particle principal axis. Results are presented for spherical, needle- and platelet-like particles at a number of different boundary layer locations along the wall-normal direction within the channel. The time evolution and probability density function of selected particle translational and rotational properties show a clear distinction between the behaviour of the various particles types, and indicate the significance of particle shape when modelling many practically relevant flows. 相似文献
39.
T. Gilliland K. K. J. Ranga-Dinesh M. Fairweather S. A. E. G. Falle K. W. Jenkins A. M. Savill 《Flow, Turbulence and Combustion》2012,89(3):385-406
Direct numerical and large eddy simulation (DNS and LES) are applied to study passive scalar mixing and intermittency in turbulent round jets. Both simulation techniques are applied to the case of a low Reynolds number jet with Re = 2,400, whilst LES is also used to predict a high Re = 68,000 flow. Comparison between time-averaged results for the scalar field of the low Re case demonstrate reasonable agreement between the DNS and LES, and with experimental data and the predictions of other authors. Scalar probability density functions (pdfs) for this jet derived from the simulations are also in reasonable accord, although the DNS results demonstrate the more rapid influence of scalar intermittency with radial distance in the jet. This is reflected in derived intermittency profiles, with LES generally giving profiles that are too broad compared to equivalent DNS results, with too low a rate of decay with radial distance. In contrast, good agreement is in general found between LES predictions and experimental data for the mixing field, scalar pdfs and external intermittency in the high Reynolds number jet. Overall, the work described indicates that improved sub-grid scale modelling for use with LES may be beneficial in improving the accuracy of external intermittency predictions by this technique over the wide range of Reynolds numbers of practical interest. 相似文献
40.