10,201 research outputs found
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
Efficient computation of partition of unity interpolants through a block-based searching technique
In this paper we propose a new efficient interpolation tool, extremely
suitable for large scattered data sets. The partition of unity method is used
and performed by blending Radial Basis Functions (RBFs) as local approximants
and using locally supported weight functions. In particular we present a new
space-partitioning data structure based on a partition of the underlying
generic domain in blocks. This approach allows us to examine only a reduced
number of blocks in the search process of the nearest neighbour points, leading
to an optimized searching routine. Complexity analysis and numerical
experiments in two- and three-dimensional interpolation support our findings.
Some applications to geometric modelling are also considered. Moreover, the
associated software package written in \textsc{Matlab} is here discussed and
made available to the scientific community
Adaptive meshless centres and RBF stencils for Poisson equation
We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method
Reconstruction of inhomogeneous metric perturbations and electromagnetic four-potential in Kerr spacetime
We present a procedure that allows the construction of the metric
perturbations and electromagnetic four-potential, for gravitational and
electromagnetic perturbations produced by sources in Kerr spacetime. This may
include, for example, the perturbations produced by a point particle or an
extended object moving in orbit around a Kerr black hole. The construction is
carried out in the frequency domain. Previously, Chrzanowski derived the vacuum
metric perturbations and electromagnetic four-potential by applying a
differential operator to a certain potential . Here we construct
for inhomogeneous perturbations, thereby allowing the application of
Chrzanowski's method. We address this problem in two stages: First, for vacuum
perturbations (i.e. pure gravitational or electromagnetic waves), we construct
the potential from the modes of the Weyl scalars or .
Second, for perturbations produced by sources, we express in terms of
the mode functions of the source, i.e. the energy-momentum tensor or the electromagnetic current vector .Comment: 20 pages; few typos corrected and minor modifications made; accepted
to Phys. Rev.
Numerical Solution of Partial Differential Equations Using Polynomial Particular Solutions
Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning.
The derived polynomial particular solutions are also applied in the localized method of particular solutions to solve large-scale problems. Many numerical experiments have been performed to show the effectiveness of the particular solutions on this algorithm.
As another part of the dissertation, a modified method of particular solutions (MPS) has been used for solving nonlinear Poisson-type problems defined on different geometries. Polyharmonic splines are used as the basis functions so that no shape parameter is needed in the solution process. The MPS is also applied to compute the sizes of critical domains of different shapes for a quenching problem. These sizes are compared with the sizes of critical domains obtained from some other numerical methods. Numerical examples are presented to show the efficiency and accuracy of the method
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