3,335 research outputs found
High order Poisson Solver for unbounded flows
AbstractThis paper presents a high order method for solving the unbounded Poisson equation on a regular mesh using a Green's function solution. The high order convergence was achieved by formulating mollified integration kernels, that were derived from a filter regularisation of the solution field. The method was implemented on a rectangular domain using fast Fourier transforms (FFT) to increase computational efficiency. The Poisson solver was extended to directly solve the derivatives of the solution. This is achieved either by including the differential operator in the integration kernel or by performing the differentiation as a multiplication of the Fourier coefficients. In this way, differential operators such as the divergence or curl of the solution field could be solved to the same high order convergence without additional computational effort. The method was applied and validated using the equations of fluid mechanics as an example, but can be used in many physical problems to solve the Poisson equation on a rectangular unbounded domain. For the two-dimensional case we propose an infinitely smooth test function which allows for arbitrary high order convergence. Using Gaussian smoothing as regularisation we document an increased convergence rate up to tenth order. The method however, can easily be extended well beyond the tenth order. To show the full extend of the method we present the special case of a spectrally ideal regularisation of the velocity formulated integration kernel, which achieves an optimal rate of convergence
Non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions
This paper is a revised version of the original paper of same
title--published in Applied Mathematics Letters 89--containing some corrections
and clarifications to the original text. We derive non-singular Green's
functions for the unbounded Poisson equation in one, two and three dimensions,
using a cut-off function in the Fourier domain to impose a smallest length
scale when deriving the Green's function. The resulting non-singular Green's
functions are relevant to applications which are restricted to a minimum
resolved length scale (e.g. a mesh size h) and thus cannot handle the singular
Green's function of the continuous Poisson equation. We furthermore derive the
gradient vector of the non-singular Green's function, as this is useful in
applications where the Poisson equation represents potential functions of a
vector field
Fast integral equation methods for the modified Helmholtz equation
We present a collection of integral equation methods for the solution to the
two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) =
0, in bounded or unbounded multiply-connected domains. We consider both
Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral
equations of the second kind, which are discretized using high-order, hybrid
Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure
requires only O(N) or operations, where N is the number of nodes
in the discretization of the boundary. We demonstrate the performance of the
methods on several numerical examples.Comment: Published in Computers & Mathematics with Application
A domain-decomposition method to implement electrostatic free boundary conditions in the radial direction for electric discharges
At high pressure electric discharges typically grow as thin, elongated
filaments. In a numerical simulation this large aspect ratio should ideally
translate into a narrow, cylindrical computational domain that envelops the
discharge as closely as possible. However, the development of the discharge is
driven by electrostatic interactions and, if the computational domain is not
wide enough, the boundary conditions imposed to the electrostatic potential on
the external boundary have a strong effect on the discharge. Most numerical
codes for electric discharges circumvent this problem by either using a wide
computational domain or by calculating the boundary conditions by integrating
the Green's function of an infinite domain. Here we describe an accurate and
efficient method to impose free boundary conditions for an elongated electric
discharge. To facilitate the use of our method we provide a sample
implementation.Comment: 21 pages, 4 figures, a movie and a sample code in python. A new
Appendix has been adde
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
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